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Cryptography and network SeCurityPrinciPles and Practice

Sixth edition

William Stallings

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Library of Congress Cataloging-in-Publication Data on file.

For Tricia never dull never boring the smartest and bravest

person I know

iii

ContentS

Notation xi

Preface xiii

Chapter 0 Guide for Readers and Instructors 1

0.1 Outline of This Book 2 0.2 A Roadmap for Readers and Instructors 3 0.3 Internet and Web Resources 4 0.4 Standards 5

Chapter 1 Overview 7

1.1 Computer Security Concepts 9 1.2 The OSI Security Architecture 14 1.3 Security Attacks 15 1.4 Security Services 17 1.5 Security Mechanisms 20 1.6 A Model for Network Security 22 1.7 Recommended Reading 24 1.8 Key Terms, Review Questions, and Problems 25

Part One Symmetric ciPherS 27

Chapter 2 Classical Encryption Techniques 27

2.1 Symmetric Cipher Model 28 2.2 Substitution Techniques 34 2.3 Transposition Techniques 49 2.4 Rotor Machines 50 2.5 Steganography 52 2.6 Recommended Reading 54 2.7 Key Terms, Review Questions, and Problems 55

Chapter 3 Block Ciphers and the Data Encryption Standard 61

3.1 Traditional Block Cipher Structure 63 3.2 The Data Encryption Standard 72 3.3 A DES Example 74 3.4 The Strength of DES 77 3.5 Block Cipher Design Principles 78 3.6 Recommended Reading 80 3.7 Key Terms, Review Questions, and Problems 81

Chapter 4 Basic Concepts in Number Theory and Finite Fields 85

4.1 Divisibility and the Division Algorithm 87 4.2 The Euclidean Algorithm 88

iv Contents

4.3 Modular Arithmetic 91 4.4 Groups, Rings, and Fields 99 4.5 Finite Fields of the Form GF( p) 102 4.6 Polynomial Arithmetic 106 4.7 Finite Fields of the Form GF(2n) 112 4.8 Recommended Reading 124 4.9 Key Terms, Review Questions, and Problems 124 Appendix 4A The Meaning of mod 127

Chapter 5 Advanced Encryption Standard 129

5.1 Finite Field Arithmetic 130 5.2 AES Structure 132 5.3 AES Transformation Functions 137 5.4 AES Key Expansion 148 5.5 An AES Example 151 5.6 AES Implementation 155 5.7 Recommended Reading 159 5.8 Key Terms, Review Questions, and Problems 160 Appendix 5A Polynomials with Coefficients in GF(28) 162 Appendix 5B Simplified AES 164

Chapter 6 Block Cipher Operation 174

6.1 Multiple Encryption and Triple DES 175 6.2 Electronic Code book 180 6.3 Cipher Block Chaining Mode 183 6.4 Cipher Feedback Mode 185 6.5 Output Feedback Mode 187 6.6 Counter Mode 189 6.7 XTS-AES Mode for Block-Oriented Storage Devices 191 6.8 Recommended Reading 198 6.9 Key Terms, Review Questions, and Problems 198

Chapter 7 Pseudorandom Number Generation and Stream Ciphers 202

7.1 Principles of Pseudorandom Number Generation 203 7.2 Pseudorandom Number Generators 210 7.3 Pseudorandom Number Generation Using a Block Cipher 213 7.4 Stream Ciphers 219 7.5 RC4 221 7.6 True Random Number Generators 223 7.7 Recommended Reading 227 7.8 Key Terms, Review Questions, and Problems 228

Part twO aSymmetric ciPherS 231

Chapter 8 More Number Theory 231

8.1 Prime Numbers 232 8.2 Fermat’s and Euler’s Theorems 236 8.3 Testing for Primality 239 8.4 The Chinese Remainder Theorem 242

Contents v

8.5 Discrete Logarithms 244 8.6 Recommended Reading 249 8.7 Key Terms, Review Questions, and Problems 250

Chapter 9 Public-Key Cryptography and RSA 253

9.1 Principles of Public-Key Cryptosystems 256 9.2 The RSA Algorithm 264 9.3 Recommended Reading 278 9.4 Key Terms, Review Questions, and Problems 279 Appendix 9A The Complexity of Algorithms 283

Chapter 10 Other Public-Key Cryptosystems 286

10.1 Diffie-Hellman Key Exchange 287 10.2 Elgamal Cryptographic System 292 10.3 Elliptic Curve Arithmetic 295 10.4 Elliptic Curve Cryptography 303 10.5 Pseudorandom Number Generation Based on an Asymmetric Cipher 306 10.6 Recommended Reading 309 10.7 Key Terms, Review Questions, and Problems 309

Part three cryPtOgraPhic Data integrity algOrithmS 313

Chapter 11 Cryptographic Hash Functions 313

11.1 Applications of Cryptographic Hash Functions 315 11.2 Two Simple Hash Functions 320 11.3 Requirements and Security 322 11.4 Hash Functions Based on Cipher Block Chaining 328 11.5 Secure Hash Algorithm (SHA) 329 11.6 SHA-3 339 11.7 Recommended Reading 351 11.8 Key Terms, Review Questions, and Problems 351

Chapter 12 Message Authentication Codes 355

12.1 Message Authentication Requirements 357 12.2 Message Authentication Functions 357 12.3 Requirements for Message Authentication Codes 365 12.4 Security of MACs 367 12.5 MACs Based on Hash Functions: HMAC 368 12.6 MACs Based on Block Ciphers: DAA and CMAC 373 12.7 Authenticated Encryption: CCM and GCM 376 12.8 Key Wrapping 382 12.9 Pseudorandom Number Generation using Hash Functions and MACs 387 12.10 Recommended Reading 390 12.11 Key Terms, Review Questions, and Problems 390

Chapter 13 Digital Signatures 393

13.1 Digital Signatures 395 13.2 Elgamal Digital Signature Scheme 398 13.3 Schnorr Digital Signature Scheme 400

vi Contents

13.4 NIST Digital Signature Algorithm 401 13.5 Elliptic Curve Digital Signature Algorithm 404 13.6 RSA-PSS Digital Signature Algorithm 407 13.7 Recommended Reading 412 13.8 Key Terms, Review Questions, and Problems 412

Part FOur mutual truSt 417

Chapter 14 Key Management and Distribution 417

14.1 Symmetric Key Distribution Using Symmetric Encryption 418 14.2 Symmetric Key Distribution Using Asymmetric Encryption 427 14.3 Distribution of Public Keys 430 14.4 X.509 Certificates 435 14.5 Public-Key Infrastructure 443 14.6 Recommended Reading 445 14.7 Key Terms, Review Questions, and Problems 446

Chapter 15 User Authentication 450

15.1 Remote User-Authentication Principles 451 15.2 Remote User-Authentication Using Symmetric Encryption 454 15.3 Kerberos 458 15.4 Remote User Authentication Using Asymmetric Encryption 476 15.5 Federated Identity Management 478 15.6 Personal Identity Verification 484 15.7 Recommended Reading 491 15.8 Key Terms, Review Questions, and Problems 491

Part Five netwOrk anD internet Security 495

Chapter 16 Network Access Control and Cloud Security 495

16.1 Network Access Control 496 16.2 Extensible Authentication Protocol 499 16.3 IEEE 802.1X Port-Based Network Access Control 503 16.4 Cloud Computing 505 16.5 Cloud Security Risks and Countermeasures 512 16.6 Data Protection in the Cloud 514 16.7 Cloud Security as a Service 517 16.8 Recommended Reading 520 16.9 Key Terms, Review Questions, and Problems 521

Chapter 17 Transport-Level Security 522

17.1 Web Security Considerations 523 17.2 Secure Sockets Layer 525 17.3 Transport Layer Security 539 17.4 HTTPS 543 17.5 Secure Shell (SSH) 544 17.6 Recommended Reading 555 17.7 Key Terms, Review Questions, and Problems 556

Contents vii

Chapter 18 Wireless Network Security 558

18.1 Wireless Security 559 18.2 Mobile Device Security 562 18.3 IEEE 802.11 Wireless LAN Overview 566 18.4 IEEE 802.11i Wireless LAN Security 572 18.5 Recommended Reading 586 18.6 Key Terms, Review Questions, and Problems 587

Chapter 19 Electronic Mail Security 590

19.1 Pretty Good Privacy 591 19.2 S/MIME 599 19.3 DomainKeys Identified Mail 615 19.4 Recommended Reading 622 19.5 Key Terms, Review Questions, and Problems 622 Appendix 19A Radix-64 Conversion 623

Chapter 20 IP Security 626

20.1 IP Security Overview 628 20.2 IP Security Policy 632 20.3 Encapsulating Security Payload 638 20.4 Combining Security Associations 645 20.5 Internet Key Exchange 649 20.6 Cryptographic Suites 657 20.7 Recommended Reading 659 20.8 Key Terms, Review Questions, and Problems 659

aPPenDiceS 661

Appendix A Projects for Teaching Cryptography and Network Security 661

A.1 Sage Computer Algebra Projects 662 A.2 Hacking Project 663 A.3 Block Cipher Projects 664 A.4 Laboratory Exercises 664 A.5 Research Projects 664 A.6 Programming Projects 665 A.7 Practical Security Assessments 665 A.8 Firewall Projects 666 A.9 Case Studies 666 A.10 Writing Assignments 666 A.11 Reading/Report Assignments 667 A.12 Discussion Topics 667

Appendix B Sage Examples 668

B.1 Linear Algebra and Matrix Functionality 669 B.2 Chapter 2: Classical Encryption 670 B.3 Chapter 3: Block Ciphers and the Data Encryption Standard 673 B.4 Chapter 4: Basic Concepts in Number Theory and Finite Fields 677 B.5 Chapter 5: Advanced Encryption Standard 684

viii Contents

B.6 Chapter 6: Pseudorandom Number Generation and Stream Ciphers 689 B.7 Chapter 8: Number Theory 691 B.8 Chapter 9: Public-Key Cryptography and RSA 696 B.9 Chapter 10: Other Public-Key Cryptosystems 699 B.10 Chapter 11: Cryptographic Hash Functions 704 B.11 Chapter 13: Digital Signatures 706

References 710

Credits 720

Index 723

Online chaPterS anD aPPenDiceS1

Part Six SyStem Security

Chapter 21 Malicious Software

21.1 Types of Malicious Software 21.2 Propagation – Infected Content - Viruses 21.3 Propagation – Vulnerability Exploit - Worms 21.4 Propagation – Social Engineering – SPAM, Trojans 21.5 Payload – System Corruption 21.6 Payload – Attack Agent – Zombie, Bots 21.7 Payload – Information Theft – Keyloggers, Phishing, Spyware 21.8 Payload – Stealthing – Backdoors, Rootkits 21.9 Countermeasures 21.10 Distributed Denial of Service Attacks 21.11 Recommended Reading 21.12 Key Terms, Review Questions, and Problems

Chapter 22 Intruders

22.1 Intruders 22.2 Intrusion Detection 22.3 Password Management 22.4 Recommended Reading 22.5 Key Terms, Review Questions, and Problems Appendix 22A The Base-Rate Fallacy

Chapter 23 Firewalls

23.1 The Need for Firewalls 23.2 Firewall Characteristics 23.3 Types of Firewalls 23.4 Firewall Basing 23.5 Firewall Location and Configurations 23.6 Recommended Reading 23.7 Key Terms, Review Questions, and Problems

1Online chapters, appendices, and other documents are Premium Content, available via the access card at the front of this book.

Contents ix

Part Seven legal anD ethical iSSueS

Chapter 24 Legal and Ethical Issues

24.1 Cybercrime and Computer Crime 24.2 Intellectual Property 24.3 Privacy 24.4 Ethical Issues 24.5 Recommended Reading 24.6 Key Terms, Review Questions, and Problems

Appendix C Sage Exercises

Appendix D Standards and Standards-Setting Organizations

Appendix E Basic Concepts from Linear Algebra

Appendix F Measures of Security and Secrecy

Appendix G Simplified DES

Appendix H Evaluation Criteria for AES

Appendix I More on Simplified AES

Appendix J Knapsack Public-Key Algorithm

Appendix K Proof of the Digital Signature Algorithm

Appendix L TCP/IP and OSI

Appendix M Java Cryptographic APIs

Appendix N MD5 and Whirlpool Hash Functions

Appendix O Data Compression Using ZIP

Appendix P More on PGP

Appendix Q The International Reference Alphabet

Appendix R Proof of the RSA Algorithm

Appendix S Data Encryption Standard (DES)

Appendix T Kerberos Encryption Techniques

Appendix U Mathematical Basis of the Birthday Attack

Appendix V Evaluation Criteria for SHA-3

Glossary


notation

Even the natives have difficulty mastering this peculiar vocabulary.

—The Golden Bough, Sir James George Frazer

Symbol Expression Meaning

D, K D(K, Y ) Symmetric decryption of ciphertext Y using secret key K

D, PRa D(PRa, Y ) Asymmetric decryption of ciphertext Y using A’s private key PRa

D, PUa D(PUa, Y ) Asymmetric decryption of ciphertext Y using A’s public key PUa

E, K E(K, X ) Symmetric encryption of plaintext X using secret key K

E, PRa E(PRa, X ) Asymmetric encryption of plaintext X using A’s private key PRa

E, PUa E(PUa, X ) Asymmetric encryption of plaintext X using A’s public key PUa

K Secret key

PRa Private key of user A

PUa Public key of user A

MAC, K MAC(K, X ) Message authentication code of message X using secret key K

GF( p) The finite field of order p, where p is prime.The field is defined as the set Zp together with the arithmetic operations modulo p.

GF(2n) The finite field of order 2n

Zn Set of nonnegative integers less than n

gcd gcd(i, j)Greatest common divisor; the largest positive integer that divides both i and j with no remainder on division.

mod a mod m Remainder after division of a by m

mod, K a K b (mod m) a mod m = b mod m

mod, [ a [ b (mod m) a mod m ≠ b mod m

dlog dloga, p(b) Discrete logarithm of the number b for the base a (mod p)

w f(n)The number of positive integers less than n and relatively prime to n.This is Euler’s totient function.

Σ an

i = 1 ai a1 + a2 + c + an

Π Πn

i = 1ai a1 * a2 * c * an

xi

xii notation

Symbol Expression Meaning

| i | j i divides j, which means that there is no remainder when j is divided by i

| , | | a | Absolute value of a

|| x || y x concatenated with y

≈ x ≈ y x is approximately equal to y

⊕ x ⊕ yExclusive-OR of x and y for single-bit variables;Bitwise exclusive-OR of x and y for multiple-bit variables

:,; :x; The largest integer less than or equal to x

∈ x ∈ S The element x is contained in the set S.

· A · (a1, a2, c ak)

The integer A corresponds to the sequence of integers (a1, a2, c ak)

prefaCe

“There is the book, Inspector. I leave it with you, and you cannot doubt that it contains a full explanation.”

—The Adventure of the Lion’s Mane, Sir Arthur Conan Doyle

what’S new in the Sixth eDitiOn

In the four years since the fifth edition of this book was published, the field has seen contin-ued innovations and improvements. In this new edition, I try to capture these changes while maintaining a broad and comprehensive coverage of the entire field. To begin this process of revision, the fifth edition of this book was extensively reviewed by a number of professors who teach the subject and by professionals working in the field. The result is that, in many places, the narrative has been clarified and tightened, and illustrations have been improved.

Beyond these refinements to improve pedagogy and user-friendliness, there have been substantive changes throughout the book. Roughly the same chapter organization has been retained, but much of the material has been revised and new material has been added. The most noteworthy changes are as follows:

• Network access control: A new chapter provides coverage of network access control, including a general overview plus discussions of the Extensible Authentication Proto-col and IEEE 802.1X.

• Cloud security: A new section covers the security issues relating to the exciting new area of cloud computing.

• SHA-3: A new section covers the new cryptographic hash standard, SHA-3, which was adopted in 2012.

• Key wrapping: The use of key wrapping to protect symmetric keys has been adopted in a number of applications. A new section covers this topic.

• Elliptic Curve Digital Signature Algorithm (ECDSA): Because ECDSA is more effi-cient than other digital signature schemes, it is increasingly being adopted for digital signature applications. A new section covers ECDSA.

• RSA Probabilistic Signature Scheme (RSA-PSS): RSA-based digital signature schemes are perhaps the most widely used. A new section covers the recently standard-ized RSA-PSS, which is in the process of replacing older RSA-based schemes.

• True random number generator: True random number generators have traditionally had a limited role because of their low bit rate, but a new generation of hardware true random number generators is now available that is comparable in performance to soft-ware pseudorandom number generators. A new section covers this topic and discusses the Intel Digital Random Number Generator (DRNG).

• Personal identity verification (PIV): The NIST has issued a comprehensive set of standards for smartcard-based user authentication that is being widely adopted. A new section covers PIV.

xiii

xiv PrefaCe

• Mobile device security: Mobile device security has become an essential aspect of enter-prise network security. A new section covers this important topic.

• Malicious software: This chapter provides a different focus than the chapter on mali-cious software in the previous edition. Increasingly we see backdoor/rootkit type mal-ware installed by social engineering attacks, rather than more classic virus/worm direct infection. And phishing is even more prominent than ever. These trends are reflected in the coverage.

• Sample syllabus: The text contains more material than can be conveniently covered in one semester. Accordingly, instructors are provided with several sample syllabi that guide the use of the text within limited time (e.g., 16 weeks or 12 weeks). These samples are based on real-world experience by professors with the fifth edition.

• VideoNotes on Sage examples: The new edition is accompanied by a number of VideoNotes lectures that amplify and clarify the cryptographic examples presented in Appendix B, which introduces Sage.

• Learning objectives: Each chapter now begins with a list of learning objectives.

ObjectiveS

It is the purpose of this book to provide a practical survey of both the principles and practice of cryptography and network security. In the first part of the book, the basic issues to be addressed by a network security capability are explored by providing a tutorial and survey of cryptography and network security technology. The latter part of the book deals with the practice of network security: practical applications that have been implemented and are in use to provide network security.

The subject, and therefore this book, draws on a variety of disciplines. In particular, it is impossible to appreciate the significance of some of the techniques discussed in this book without a basic understanding of number theory and some results from probability theory. Nevertheless, an attempt has been made to make the book self-contained. The book not only presents the basic mathematical results that are needed but provides the reader with an intuitive understanding of those results. Such background material is introduced as needed. This approach helps to motivate the material that is introduced, and the author considers this preferable to simply presenting all of the mathematical material in a lump at the begin-ning of the book.

SuPPOrt OF acm/ieee cOmPuter Science curricula 2013

The book is intended for both academic and professional audiences. As a textbook, it is intended as a one-semester undergraduate course in cryptography and network security for computer science, computer engineering, and electrical engineering majors. The changes to this edition are intended to provide support of the current draft version of the ACM/IEEE Computer Science Curricula 2013 (CS2013). CS2013 adds Information Assurance and Security (IAS) to the curriculum recommendation as one of the Knowledge Areas in the Computer Science Body of Knowledge. The document states that IAS is now part of the curriculum recommendation because of the critical role of IAS in computer science educa-tion. CS2013 divides all course work into three categories: Core-Tier 1 (all topics should be included in the curriculum), Core-Tier-2 (all or almost all topics should be included), and

PrefaCe xv

elective (desirable to provide breadth and depth). In the IAS area, CS2013 recommends topics in Fundamental Concepts and Network Security in Tier 1 and Tier 2, and Cryptog-raphy topics as elective. This text covers virtually all of the topics listed by CS2013 in these three categories.

The book also serves as a basic reference volume and is suitable for self-study.

Plan OF the text

The book is divided into seven parts, which are described in Chapter 0.

• Symmetric Ciphers

• Asymmetric Ciphers

• Cryptographic Data Integrity Algorithms

• Mutual Trust

• Network and Internet Security

• System Security

• Legal and Ethical Issues

The book includes a number of pedagogic features, including the use of the computer algebra system Sage and numerous figures and tables to clarify the discussions. Each chapter includes a list of key words, review questions, homework problems, and suggestions for further reading. The book also includes an extensive glossary, a list of frequently used acronyms, and a bibliography. In addition, a test bank is available to instructors.

inStructOr SuPPOrt materialS

The major goal of this text is to make it as effective a teaching tool for this exciting and fast-moving subject as possible. This goal is reflected both in the structure of the book and in the supporting material. The text is accompanied by the following supplementary material that will aid the instructor:

• Solutions manual: Solutions to all end-of-chapter Review Questions and Problems.

• Projects manual: Suggested project assignments for all of the project categories listed below.

• PowerPoint slides: A set of slides covering all chapters, suitable for use in lecturing.

• PDF files: Reproductions of all figures and tables from the book.

• Test bank: A chapter-by-chapter set of questions with a separate file of answers.

• Sample syllabuses: The text contains more material than can be conveniently covered in one semester. Accordingly, instructors are provided with several sample syllabuses that guide the use of the text within limited time. These samples are based on real-world experience by professors with the fifth edition.

All of these support materials are available at the Instructor Resource Center (IRC) for this textbook, which can be reached through the publisher’s Web site www.pearsonhighered .com/stallings or by clicking on the link labeled Pearson Resources for Instructors at this book’s

www.pearsonhighered.com/stallings

www.pearsonhighered.com/stallings

xvi PrefaCe

Companion Web site at WilliamStallings.com/Cryptography. To gain access to the IRC, please contact your local Pearson sales representative via pearsonhighered.com/educator/replocator/requestSalesRep.page or call Pearson Faculty Services at 1-800-526-0485.

The Companion Web site, at WilliamStallings.com/Cryptography (click on Instructor Resources link), includes the following:

• Links to Web sites for other courses being taught using this book

• Sign-up information for an Internet mailing list for instructors using this book to exchange information, suggestions, and questions with each other and with the author

PrOjectS anD Other StuDent exerciSeS

For many instructors, an important component of a cryptography or network security course is a project or set of projects by which the student gets hands-on experience to reinforce concepts from the text. This book provides an unparalleled degree of support, including a projects component in the course. The IRC not only includes guidance on how to assign and structure the projects, but also includes a set of project assignments that covers a broad range of topics from the text:

• Sage projects: Described in the next section.

• Hacking project: Exercise designed to illuminate the key issues in intrusion detection and prevention.

• Block cipher projects: A lab that explores the operation of the AES encryption algo-rithm by tracing its execution, computing one round by hand, and then exploring the various block cipher modes of use. The lab also covers DES. In both cases, an online Java applet is used (or can be downloaded) to execute AES or DES.

• Lab exercises: A series of projects that involve programming and experimenting with concepts from the book.

• Research projects: A series of research assignments that instruct the student to research a particular topic on the Internet and write a report.

• Programming projects: A series of programming projects that cover a broad range of topics and that can be implemented in any suitable language on any platform.

• Practical security assessments: A set of exercises to examine current infrastructure and practices of an existing organization.

• Firewall projects: A portable network firewall visualization simulator, together with exercises for teaching the fundamentals of firewalls.

• Case studies: A set of real-world case studies, including learning objectives, case description, and a series of case discussion questions.

• Writing assignments: A set of suggested writing assignments, organized by chapter.

• Reading/report assignments: A list of papers in the literature—one for each chapter—that can be assigned for the student to read and then write a short report.

This diverse set of projects and other student exercises enables the instructor to use the book as one component in a rich and varied learning experience and to tailor a course plan to meet the specific needs of the instructor and students. See Appendix A in this book for details.

PrefaCe xvii

the Sage cOmPuter algebra SyStem

One of the most important features of this book is the use of Sage for cryptographic exam-ples and homework assignments. Sage is an open-source, multiplatform, freeware package that implements a very powerful, flexible, and easily learned mathematics and computer algebra system. Unlike competing systems (such as Mathematica, Maple, and MATLAB), there are no licensing agreements or fees involved. Thus, Sage can be made available on computers and networks at school, and students can individually download the software to their own personal computers for use at home. Another advantage of using Sage is that students learn a powerful, flexible tool that can be used for virtually any mathematical application, not just cryptography.

The use of Sage can make a significant difference to the teaching of the mathematics of cryptographic algorithms. This book provides a large number of examples of the use of Sage covering many cryptographic concepts in Appendix B, which is included in this book.

Appendix C lists exercises in each of these topic areas to enable the student to gain hands-on experience with cryptographic algorithms. This appendix is available to instruc-tors at the IRC for this book. Appendix C includes a section on how to download and get started with Sage, a section on programming with Sage, and exercises that can be assigned to students in the following categories:

• Chapter 2—Classical Encryption: Affine ciphers and the Hill cipher.

• Chapter 3—Block Ciphers and the Data Encryption Standard: Exercises based on SDES.

• Chapter 4—Basic Concepts in Number Theory and Finite Fields: Euclidean and extended Euclidean algorithms, polynomial arithmetic, and GF(24).

• Chapter 5—Advanced Encryption Standard: Exercises based on SAES.

• Chapter 6—Pseudorandom Number Generation and Stream Ciphers: Blum Blum Shub, linear congruential generator, and ANSI X9.17 PRNG.

• Chapter 8—Number Theory: Euler’s Totient function, Miller Rabin, factoring, modu-lar exponentiation, discrete logarithm, and Chinese remainder theorem.

• Chapter 9—Public-Key Cryptography and RSA: RSA encrypt/decrypt and signing.

• Chapter 10—Other Public-Key Cryptosystems: Diffie-Hellman, elliptic curve.

• Chapter 11—Cryptographic Hash Functions: Number-theoretic hash function.

• Chapter 13—Digital Signatures: DSA.

Online DOcumentS FOr StuDentS

For this new edition, a tremendous amount of original supporting material for students has been made available online, at two Web locations. The Companion Web site, at WilliamStallings.com/Cryptography (click on Student Resources link), includes a list of rel-evant links organized by chapter and an errata sheet for the book.

Purchasing this textbook new also grants the reader six months of access to the Premium Content site, which includes the following materials:

• Online chapters: To limit the size and cost of the book, four chapters of the book are provided in PDF format. This includes three chapters on computer security

xviii PrefaCe

and one on legal and ethical issues. The chapters are listed in this book’s table of contents.

• Online appendices: There are numerous interesting topics that support material found in the text but whose inclusion is not warranted in the printed text. A total of 20 online appendices cover these topics for the interested student. The appendices are listed in this book’s table of contents.

• Homework problems and solutions: To aid the student in understanding the material, a separate set of homework problems with solutions are available.

• Key papers: A number of papers from the professional literature, many hard to find, are provided for further reading.

• Supporting documents: A variety of other useful documents are referenced in the text and provided online.

• Sage code: The Sage code from the examples in Appendix B is useful in case the student wants to play around with the examples.

To access the Premium Content site, click on the Premium Content link at the Com-panion Web site or at pearsonhighered.com/stallings and enter the student access code found on the card in the front of the book.

acknOwleDgmentS

This new edition has benefited from review by a number of people who gave generously of their time and expertise. The following people reviewed all or a large part of the manuscript: Steven Tate (University of North Carolina at Greensboro), Kemal Akkaya (Southern Illinois University), Bulent Yener (Rensselaer Polytechnic Institute), Ellen Gethner (University of Colorado, Denver), Stefan A. Robila (Montclair State University), and Albert Levi (Sabanci University, Istanbul, Turkey).

Thanks also to the people who provided detailed technical reviews of one or more chapters: Kashif Aftab, Jon Baumgardner, Alan Cantrell, Rajiv Dasmohapatra, Edip Demirbilek, Dhananjoy Dey, Dan Dieterle, Gerardo Iglesias Galvan, Michel Garcia, David Gueguen, Anasuya Threse Innocent, Dennis Kavanagh, Duncan Keir, Robert Knox, Bob Kupperstein, Bo Lin, Kousik Nandy, Nickolay Olshevsky, Massimiliano Sembiante, Oscar So, and Varun Tewari.

In addition, I was fortunate to have reviews of individual topics by “subject-area gurus,” including Jesse Walker of Intel (Intel’s Digital Random Number Generator), Russ Housley of Vigil Security (key wrapping), Joan Daemen (AES), Edward F. Schaefer of Santa Clara University (Simplified AES), Tim Mathews, formerly of RSA Laboratories (S/MIME), Alfred Menezes of the University of Waterloo (elliptic curve cryptography), William Sutton, Editor/Publisher of The Cryptogram (classical encryption), Avi Rubin of Johns Hopkins University (number theory), Michael Markowitz of Information Security Corporation (SHA and DSS), Don Davis of IBM Internet Security Systems (Kerberos), Steve Kent of BBN Technologies (X.509), and Phil Zimmerman (PGP).

Nikhil Bhargava (IIT Delhi) developed the set of online homework problems and so-lutions. Dan Shumow of Microsoft and the University of Washington developed all of the Sage examples and assignments in Appendices B and C. Professor Sreekanth Malladi of

PrefaCe xix

Dakota State University developed the hacking exercises. Lawrie Brown of the Australian Defence Force Academy provided the AES/DES block cipher projects and the security assessment assignments.

Sanjay Rao and Ruben Torres of Purdue University developed the laboratory exer-cises that appear in the IRC. The following people contributed project assignments that appear in the instructor’s supplement: Henning Schulzrinne (Columbia University); Cetin Kaya Koc (Oregon State University); and David Balenson (Trusted Information Systems and George Washington University). Kim McLaughlin developed the test bank.

Finally, I thank the many people responsible for the publication of this book, all of whom did their usual excellent job. This includes the staff at Pearson, particularly my editor Tracy Johnson, associate editor Carole Snyder, production supervisor Robert Engelhardt, and production project manager Pat Brown. I also thank Shiny Rajesh and the production staff at Integra for another excellent and rapid job. Thanks also to the marketing and sales staffs at Pearson, without whose efforts this book would not be in front of you.

With all this assistance, little remains for which I can take full credit. However, I am proud to say that, with no help whatsoever, I selected all of the quotations.

abOut the authOr

Dr. William Stallings has authored 17 titles, and counting revised editions, over 40 books on computer security, computer networking, and computer architecture. His writings have appeared in numerous publications, including the Proceedings of the IEEE, ACM Comput-ing Reviews and Cryptologia.

He has 11 times received the award for the best Computer Science textbook of the year from the Text and Academic Authors Association.

In over 30 years in the field, he has been a technical contributor, technical manager, and an executive with several high-technology firms. He has designed and implemented both TCP/IP-based and OSI-based protocol suites on a variety of computers and operating systems, ranging from microcomputers to mainframes. As a consultant, he has advised gov-ernment agencies, computer and software vendors, and major users on the design, selection, and use of networking software and products.

He created and maintains the Computer Science Student Resource Site at ComputerScienceStudent.com. This site provides documents and links on a variety of sub-jects of general interest to computer science students (and professionals). He is a member of the editorial board of Cryptologia, a scholarly journal devoted to all aspects of cryptology.

Dr. Stallings holds a PhD from MIT in computer science and a BS from Notre Dame in electrical engineering.


0.1 Outline of This Book

0.2 A Roadmap for Readers and Instructors

Subject MatterTopic Ordering

0.3 Internet and Web Resources

Web Sites for This BookComputer Science Student Resource SiteOther Web Sites

0.4 Standards

Chapter

Guide for readers and instructors

1

2 Chapter 0 / Guide for readers and instruCtors

The art of war teaches us to rely not on the likelihood of the enemy’s not coming, but on our own readiness to receive him; not on the chance of his not attacking, but rather on the fact that we have made our position unassailable.

—The Art of War, Sun Tzu

This book, with its accompanying Web sites, covers a lot of material. Here we give the reader an overview.

0.1 Outline Of this BOOk

Following an introductory chapter, Chapter 1, the book is organized into seven parts:

Part One: Symmetric Ciphers: Provides a survey of symmetric encryption, including classical and modern algorithms. The emphasis is on the most important algorithm, the Advanced Encryption Standard (AES). Also covered is the Data Encryption Standard (DES). This part also covers the most important stream encryption algorithm, RC4, and the topic of pseudorandom and random number generation.

Part Two: Asymmetric Ciphers: Provides a survey of public-key algorithms, including RSA (Rivest-Shamir-Adelman) and elliptic curve.

Part Three: Cryptographic Data Integrity Algorithms: Begins with a survey of cryptographic hash functions. This part then covers two approaches to data integrity that rely on cryptographic hash functions: message authentication codes and digital signatures.

Part Four: Mutual Trust: Covers key management and key distribution topics and then covers user authentication techniques.

Part Five: Network Security and Internet Security: Examines the use of crypto-graphic algorithms and security protocols to provide security over net-works and the Internet. Topics covered include network access control, cloud security, transport-level security, wireless network security, e-mail security, and IP security.

Part Six: System Security: Deals with security facilities designed to protect a computer system from security threats, including intruders, viruses, and worms. This part also looks at firewall technology.

Part Seven: Legal and Ethical Issues: Deals with the legal and ethical issues related to computer and network security.

A number of online appendices at this book’s Premium Content Web site cover additional topics relevant to the book.

0.2 / a roadmap for readers and instruCtors 3

0.2 A ROAdmAp fOR ReAdeRs And instRuctORs

Subject Matter

The material in this book is organized into four broad categories:

• Cryptographic algorithms: This is the study of techniques for ensuring the secrecy and/or authenticity of information. The three main areas of study in this category are (1) symmetric encryption, (2) asymmetric encryption, and (3) cryptographic hash functions, with the related topics of message authenti-cation codes and digital signatures.

• Mutual trust: This is the study of techniques and algorithms for providing mutual trust in two main areas. First, key management and distribution deals with establishing trust in the encryption keys used between two communicat-ing entities. Second, user authentication deals with establishing trust in the identity of a communicating partner.

• Network security: This area covers the use of cryptographic algorithms in network protocols and network applications.

• Computer security: In this book, we use this term to refer to the security of computers against intruders (e.g., hackers) and malicious software (e.g., viruses). Typically, the computer to be secured is attached to a network, and the bulk of the threats arise from the network.

The first two parts of the book deal with two distinct cryptographic approaches: symmetric cryptographic algorithms and public-key, or asymmetric, cryptographic algorithms. Symmetric algorithms make use of a single key shared by two parties. Public-key algorithms make use of two keys: a private key known only to one party and a public key available to other parties.

Topic Ordering

This book covers a lot of material. For the instructor or reader who wishes a shorter treatment, there are a number of opportunities.

To thoroughly cover the material in the first three parts, the chapters should be read in sequence. With the exception of the Advanced Encryption Standard (AES), none of the material in Part One requires any special mathematical back-ground. To understand AES, it is necessary to have some understanding of finite fields. In turn, an understanding of finite fields requires a basic background in prime numbers and modular arithmetic. Accordingly, Chapter 4 covers all of these mathematical preliminaries just prior to their use in Chapter 5 on AES. Thus, if Chapter 5 is skipped, it is safe to skip Chapter 4 as well.

Chapter 2 introduces some concepts that are useful in later chapters of Part One. However, for the reader whose sole interest is contemporary cryptography, this chapter can be quickly skimmed. The two most important symmetric cryptographic algorithms are DES and AES, which are covered in Chapters 3 and 5, respectively.


Chapter 6 covers specific techniques for using what are known as block symmetric ciphers. Chapter 7 covers stream ciphers and random number generation. These two chapters may be skipped on an initial reading, but this material is referenced in later parts of the book.

For Part Two, the only additional mathematical background that is needed is in the area of number theory, which is covered in Chapter 8. The reader who has skipped Chapters 4 and 5 should first review the material on Sections 4.1 through 4.3.

The two most widely used general-purpose public-key algorithms are RSA and elliptic curve, with RSA enjoying wider acceptance. The reader may wish to skip the material on elliptic curve cryptography in Chapter 10, at least on a first reading.

In Part Three, the topics of Sections 12.6 and 12.7 are of lesser importance.Parts Four, Five, and Six are relatively independent of each other and can be

read in any order. These three parts assume a basic understanding of the material in Parts One, Two, and Three. The five chapters of Part Five, on network and Internet security, are relatively independent of one another and can be read in any order.

0.3 inteRnet And WeB ResOuRces

There are a number of resources available on the Internet and the Web that support this book and help readers keep up with developments in this field.

Web Sites for This Book

Three Web sites provide additional resources for students and instructors.There is a Companion Web site for this book at http://williamstallings.com/

Cryptography. For students, this Web site includes a list of relevant links, organized by chapter, and an errata list for the book. For instructors, this Web site provides links to course pages by professors teaching from this book.

There is also an access-controlled Premium Content Web site, which provides a wealth of supporting material, including additional online chapters, additional on-line appendices, a set of homework problems with solutions, copies of a number of key papers in this field, and a number of other supporting documents. See the card at the front of this book for access information.

Finally, additional material for instructors, including a solutions manual and a projects manual, is available at the Instructor Resource Center (IRC) for this book. See Preface for details and access information.

Computer Science Student Resource Site

I also maintain the Computer Science Student Resource Site, at Computer ScienceStudent.com. The purpose of this site is to provide documents, information, and links for computer science students and professionals. Links and documents are organized into seven categories:

• Math: Includes a basic math refresher, a queuing analysis primer, a number system primer, and links to numerous math sites.

http://williamstallings.com/

0.4 / standards 5

• How-to: Advice and guidance for solving homework problems, writing techni-cal reports, and preparing technical presentations.

• Research resources: Links to important collections of papers, technical reports, and bibliographies.

• Other useful: A variety of other useful documents and links.

• Computer science careers: Useful links and documents for those considering a career in computer science.

• Writing help: Help in becoming a clearer, more effective writer.

• Miscellaneous topics and humor: You have to take your mind off your work once in a while.

Other Web Sites

Numerous Web sites provide information related to the topics of this book. The Companion Web site provides links to these sites, organized by chapter. In addition, there are a number of forums dealing with cryptography available on the Internet. Links to these forums are provided at the Companion Website.

0.4 stAndARds

Many of the security techniques and applications described in this book have been specified as standards. Additionally, standards have been developed to cover man-agement practices and the overall architecture of security mechanisms and services. Throughout this book, we describe the most important standards in use or being developed for various aspects of cryptography and network security. Various orga-nizations have been involved in the development or promotion of these standards. The most important (in the current context) of these organizations are as follows:

• National Institute of Standards and Technology (NIST): NIST is a U.S. fed-eral agency that deals with measurement science, standards, and technology related to U.S. government use and to the promotion of U.S. private-sector innovation. Despite its national scope, NIST Federal Information Processing Standards (FIPS) and Special Publications (SP) have a worldwide impact.

• Internet Society (ISOC): ISOC is a professional membership society with worldwide organizational and individual membership. It provides leader-ship in addressing issues that confront the future of the Internet and is the organization home for the groups responsible for Internet infrastructure standards, including the Internet Engineering Task Force (IETF) and the Internet Architecture Board (IAB). These organizations develop Internet standards and related specifications, all of which are published as Requests for Comments (RFCs).

• ITU-T: The International Telecommunication Union (ITU) is an international organization within the United Nations System in which governments and the private sector coordinate global telecom networks and services. The ITU


Telecommunication Standardization Sector (ITU-T) is one of the three sectors of the ITU. ITU-T’s mission is the production of standards covering all fields of telecommunications. ITU-T standards are referred to as Recommendations.

• ISO: The International Organization for Standardization (ISO)1 is a world-wide federation of national standards bodies from more than 140 countries, one from each country. ISO is a nongovernmental organization that pro-motes the development of standardization and related activities with a view to facilitating the international exchange of goods and services and to devel-oping cooperation in the spheres of intellectual, scientific, technological, and economic activity. ISO’s work results in international agreements that are published as International Standards.

A more detailed discussion of these organizations is contained in Appendix D.

1ISO is not an acronym (in which case it would be IOS), but it is a word, derived from the Greek, meaning equal.

7

1.1 Computer Security Concepts

A Definition of Computer SecurityExamplesThe Challenges of Computer Security

1.2 The OSI Security Architecture

1.3 Security Attacks

Passive AttacksActive Attacks

1.4 Security Services

AuthenticationAccess ControlData ConfidentialityData IntegrityNonrepudiationAvailability Service

1.5 Security Mechanisms

1.6 A Model for Network Security

1.7 Recommended Reading

1.8 Key Terms, Review Questions, and Problems

Chapter

Overview

8 Chapter 1 / Overview

The combination of space, time, and strength that must be considered as the basic elements of this theory of defense makes this a fairly complicated matter. Consequently, it is not easy to find a fixed point of departure.

—On War, Carl Von Clausewitz

Learning Objectives

After studying this chapter, you should be able to:

uDescribe the key security requirements of confidentiality, integrity, and availability.

u Discuss the types of security threats and attacks that must be dealt with and give examples of the types of threats and attacks that apply to different categories of computer and network assets.

uSummarize the functional requirements for computer security.

uDescribe the X.800 security architecture for OSI.

This book focuses on two broad areas: cryptographic algorithms and protocols, which have a broad range of applications; and network and Internet security, which rely heavily on cryptographic techniques.

Cryptographic algorithms and protocols can be grouped into four main areas:

• Symmetric encryption: Used to conceal the contents of blocks or streams of data of any size, including messages, files, encryption keys, and passwords.

• Asymmetric encryption: Used to conceal small blocks of data, such as encryp-tion keys and hash function values, which are used in digital signatures.

• Data integrity algorithms: Used to protect blocks of data, such as messages, from alteration.

• Authentication protocols: These are schemes based on the use of crypto-graphic algorithms designed to authenticate the identity of entities.

The field of network and Internet security consists of measures to deter, prevent, detect, and correct security violations that involve the transmission of information. That is a broad statement that covers a host of possibilities. To give you a feel for the areas covered in this book, consider the following examples of security violations:

1. User A transmits a file to user B. The file contains sensitive information (e.g., payroll records) that is to be protected from disclosure. User C, who is not authorized to read the file, is able to monitor the transmission and capture a copy of the file during its transmission.

2. A network manager, D, transmits a message to a computer, E, under its man-agement. The message instructs computer E to update an authorization file to include the identities of a number of new users who are to be given access to

1.1 / COmputer SeCurity COnCeptS 9

that computer. User F intercepts the message, alters its contents to add or delete entries, and then forwards the message to computer E, which accepts the message as coming from manager D and updates its authorization file accordingly.

3. Rather than intercept a message, user F constructs its own message with the desired entries and transmits that message to computer E as if it had come from manager D. Computer E accepts the message as coming from manager D and updates its authorization file accordingly.

4. An employee is fired without warning. The personnel manager sends a message to a server system to invalidate the employee’s account. When the invalidation is accomplished, the server is to post a notice to the employee’s file as confirmation of the action. The employee is able to intercept the mes-sage and delay it long enough to make a final access to the server to retrieve sensitive information. The message is then forwarded, the action taken, and the confirmation posted. The employee’s action may go unnoticed for some considerable time.

5. A message is sent from a customer to a stockbroker with instructions for various transactions. Subsequently, the investments lose value and the customer denies sending the message.

Although this list by no means exhausts the possible types of network security viola-tions, it illustrates the range of concerns of network security.

1.1 cOmputer security cOncepts

A Definition of Computer Security

The NIST Computer Security Handbook [NIST95] defines the term computer secu-rity as follows:

1RFC 4949 defines information as “facts and ideas, which can be represented (encoded) as various forms of data,” and data as “information in a specific physical representation, usually a sequence of symbols that have meaning; especially a representation of information that can be processed or produced by a computer.” Security literature typically does not make much of a distinction, nor does this book.

Computer Security: The protection afforded to an automated information system in order to attain the applicable objectives of preserving the integrity, availability, and confidentiality of information system resources (includes hardware, software, firmware, information/data, and telecommunications).

This definition introduces three key objectives that are at the heart of computer security:

• Confidentiality: This term covers two related concepts:

Data1 confidentiality: Assures that private or confidential information is not made available or disclosed to unauthorized individuals.


Privacy: Assures that individuals control or influence what information related to them may be collected and stored and by whom and to whom that information may be disclosed.

• Integrity: This term covers two related concepts:

Data integrity: Assures that information and programs are changed only in a specified and authorized manner.

System integrity: Assures that a system performs its intended function in an unimpaired manner, free from deliberate or inadvertent unauthorized manipulation of the system.

• Availability: Assures that systems work promptly and service is not denied to authorized users.

These three concepts form what is often referred to as the CIA triad. The three concepts embody the fundamental security objectives for both data and for informa-tion and computing services. For example, the NIST standard FIPS 199 (Standards for Security Categorization of Federal Information and Information Systems) lists confidentiality, integrity, and availability as the three security objectives for infor-mation and for information systems. FIPS 199 provides a useful characterization of these three objectives in terms of requirements and the definition of a loss of security in each category:

• Confidentiality: Preserving authorized restrictions on information access and disclosure, including means for protecting personal privacy and propri-etary information. A loss of confidentiality is the unauthorized disclosure of information.

• Integrity: Guarding against improper information modification or destruc-tion, including ensuring information nonrepudiation and authenticity. A loss of integrity is the unauthorized modification or destruction of information.

• Availability: Ensuring timely and reliable access to and use of information. A loss of availability is the disruption of access to or use of information or an information system.

Although the use of the CIA triad to define security objectives is well estab-lished, some in the security field feel that additional concepts are needed to present a complete picture. Two of the most commonly mentioned are as follows:

• Authenticity: The property of being genuine and being able to be verified and trusted; confidence in the validity of a transmission, a message, or message originator. This means verifying that users are who they say they are and that each input arriving at the system came from a trusted source.

• Accountability: The security goal that generates the requirement for actions of an entity to be traced uniquely to that entity. This supports nonrepudia-tion, deterrence, fault isolation, intrusion detection and prevention, and after-action recovery and legal action. Because truly secure systems are not yet an achievable goal, we must be able to trace a security breach to a responsible party. Systems must keep records of their activities to permit later forensic analysis to trace security breaches or to aid in transaction disputes.


Examples

We now provide some examples of applications that illustrate the requirements just enumerated.2 For these examples, we use three levels of impact on organizations or individuals should there be a breach of security (i.e., a loss of confidentiality, integ-rity, or availability). These levels are defined in FIPS PUB 199:

• Low: The loss could be expected to have a limited adverse effect on organi-zational operations, organizational assets, or individuals. A limited adverse effect means that, for example, the loss of confidentiality, integrity, or avail-ability might (i) cause a degradation in mission capability to an extent and duration that the organization is able to perform its primary functions, but the effectiveness of the functions is noticeably reduced; (ii) result in minor dam-age to organizational assets; (iii) result in minor financial loss; or (iv) result in minor harm to individuals.

• Moderate: The loss could be expected to have a serious adverse effect on organizational operations, organizational assets, or individuals. A serious adverse effect means that, for example, the loss might (i) cause a signifi-cant degradation in mission capability to an extent and duration that the organization is able to perform its primary functions, but the effectiveness of the functions is significantly reduced; (ii) result in significant damage to organizational assets; (iii) result in significant financial loss; or (iv) result in significant harm to individuals that does not involve loss of life or serious, life-threatening injuries.

• High: The loss could be expected to have a severe or catastrophic adverse effect on organizational operations, organizational assets, or individuals. A severe or catastrophic adverse effect means that, for example, the loss might (i) cause a severe degradation in or loss of mission capability to an extent and duration that the organization is not able to perform one or more of its pri-mary functions; (ii) result in major damage to organizational assets; (iii) result in major financial loss; or (iv) result in severe or catastrophic harm to individu-als involving loss of life or serious, life-threatening injuries.

Confidentiality Student grade information is an asset whose confidentiality is considered to be highly important by students. In the United States, the release of such information is regulated by the Family Educational Rights and Privacy Act (FERPA). Grade information should only be available to students, their parents, and employees that require the information to do their job. Student enrollment information may have a moderate confidentiality rating. While still covered by FERPA, this information is seen by more people on a daily basis, is less likely to be targeted than grade information, and results in less damage if disclosed. Directory information, such as lists of students or faculty or departmental lists, may be assigned a low confidentiality rating or indeed no rating. This information is typi-cally freely available to the public and published on a school’s Web site.

2These examples are taken from a security policy document published by the Information Technology Security and Privacy Office at Purdue University.


integrity Several aspects of integrity are illustrated by the example of a hospital patient’s allergy information stored in a database. The doctor should be able to trust that the information is correct and current. Now suppose that an employee (e.g., a nurse) who is authorized to view and update this information deliberately falsifies the data to cause harm to the hospital. The database needs to be restored to a trusted basis quickly, and it should be possible to trace the error back to the person responsible. Patient allergy information is an example of an asset with a high requirement for integrity. Inaccurate information could result in serious harm or death to a patient and expose the hospital to massive liability.

An example of an asset that may be assigned a moderate level of integrity requirement is a Web site that offers a forum to registered users to discuss some specific topic. Either a registered user or a hacker could falsify some entries or deface the Web site. If the forum exists only for the enjoyment of the users, brings in little or no advertising revenue, and is not used for something important such as research, then potential damage is not severe. The Web master may experience some data, financial, and time loss.

An example of a low integrity requirement is an anonymous online poll. Many Web sites, such as news organizations, offer these polls to their users with very few safeguards. However, the inaccuracy and unscientific nature of such polls is well understood.

availability The more critical a component or service, the higher is the level of availability required. Consider a system that provides authentication ser-vices for critical systems, applications, and devices. An interruption of service results in the inability for customers to access computing resources and staff to access the resources they need to perform critical tasks. The loss of the service translates into a large financial loss in lost employee productivity and potential customer loss.

An example of an asset that would typically be rated as having a moderate availability requirement is a public Web site for a university; the Web site provides information for current and prospective students and donors. Such a site is not a critical component of the university’s information system, but its unavailability will cause some embarrassment.

An online telephone directory lookup application would be classified as a low availability requirement. Although the temporary loss of the application may be an annoyance, there are other ways to access the information, such as a hardcopy directory or the operator.

The Challenges of Computer Security

Computer and network security is both fascinating and complex. Some of the reasons follow:

1. Security is not as simple as it might first appear to the novice. The require-ments seem to be straightforward; indeed, most of the major requirements for security services can be given self-explanatory, one-word labels: confiden-tiality, authentication, nonrepudiation, or integrity. But the mechanisms used


to meet those requirements can be quite complex, and understanding them may involve rather subtle reasoning.

2. In developing a particular security mechanism or algorithm, one must always consider potential attacks on those security features. In many cases, successful attacks are designed by looking at the problem in a completely different way, therefore exploiting an unexpected weakness in the mechanism.

3. Because of point 2, the procedures used to provide particular services are often counterintuitive. Typically, a security mechanism is complex, and it is not obvious from the statement of a particular requirement that such elabo-rate measures are needed. It is only when the various aspects of the threat are considered that elaborate security mechanisms make sense.

4. Having designed various security mechanisms, it is necessary to decide where to use them. This is true both in terms of physical placement (e.g., at what points in a network are certain security mechanisms needed) and in a logical sense (e.g., at what layer or layers of an architecture such as TCP/IP [Transmission Control Protocol/Internet Protocol] should mechanisms be placed).

5. Security mechanisms typically involve more than a particular algorithm or protocol. They also require that participants be in possession of some secret information (e.g., an encryption key), which raises questions about the cre-ation, distribution, and protection of that secret information. There also may be a reliance on communications protocols whose behavior may complicate the task of developing the security mechanism. For example, if the proper functioning of the security mechanism requires setting time limits on the tran-sit time of a message from sender to receiver, then any protocol or network that introduces variable, unpredictable delays may render such time limits meaningless.

6. Computer and network security is essentially a battle of wits between a per-petrator who tries to find holes and the designer or administrator who tries to close them. The great advantage that the attacker has is that he or she need only find a single weakness, while the designer must find and eliminate all weaknesses to achieve perfect security.

7. There is a natural tendency on the part of users and system managers to per-ceive little benefit from security investment until a security failure occurs.

8. Security requires regular, even constant, monitoring, and this is difficult in today’s short-term, overloaded environment.

9. Security is still too often an afterthought to be incorporated into a system after the design is complete rather than being an integral part of the design process.

10. Many users and even security administrators view strong security as an impedi-ment to efficient and user-friendly operation of an information system or use of information.

The difficulties just enumerated will be encountered in numerous ways as we examine the various security threats and mechanisms throughout this book.


1.2 the Osi security architecture

To assess effectively the security needs of an organization and to evaluate and choose various security products and policies, the manager responsible for security needs some systematic way of defining the requirements for security and character-izing the approaches to satisfying those requirements. This is difficult enough in a centralized data processing environment; with the use of local and wide area net-works, the problems are compounded.

ITU-T3 Recommendation X.800, Security Architecture for OSI, defines such a systematic approach.4 The OSI security architecture is useful to managers as a way of organizing the task of providing security. Furthermore, because this architecture was developed as an international standard, computer and communications vendors have developed security features for their products and services that relate to this structured definition of services and mechanisms.

For our purposes, the OSI security architecture provides a useful, if abstract, overview of many of the concepts that this book deals with. The OSI security archi-tecture focuses on security attacks, mechanisms, and services. These can be defined briefly as

• Security attack: Any action that compromises the security of information owned by an organization.

• Security mechanism: A process (or a device incorporating such a process) that is designed to detect, prevent, or recover from a security attack.

• Security service: A processing or communication service that enhances the security of the data processing systems and the information transfers of an organization. The services are intended to counter security attacks, and they make use of one or more security mechanisms to provide the service.

In the literature, the terms threat and attack are commonly used to mean more or less the same thing. Table 1.1 provides definitions taken from RFC 4949, Internet Security Glossary.

3The International Telecommunication Union (ITU) Telecommunication Standardization Sector (ITU-T) is a United Nations-sponsored agency that develops standards, called Recommendations, relating to telecommunications and to open systems interconnection (OSI).4The OSI security architecture was developed in the context of the OSI protocol architecture, which is described in Appendix L. However, for our purposes in this chapter, an understanding of the OSI proto-col architecture is not required.

Table 1.1 Threats and Attacks (RFC 4949)

ThreatA potential for violation of security, which exists when there is a circumstance, capability, action, or event that could breach security and cause harm. That is, a threat is a possible danger that might exploit a vulnerability.

AttackAn assault on system security that derives from an intelligent threat; that is, an intelligent act that is a deliberate attempt (especially in the sense of a method or technique) to evade security services and violate the security policy of a system.

1.3 / SeCurity attaCkS 15

1.3 security attacks

A useful means of classifying security attacks, used both in X.800 and RFC 4949, is in terms of passive attacks and active attacks (Figure 1.1). A passive attack attempts to learn or make use of information from the system but does not affect system resources. An active attack attempts to alter system resources or affect their operation.

Passive Attacks

Passive attacks (Figure 1.1) are in the nature of eavesdropping on, or monitoring of, transmissions. The goal of the opponent is to obtain information that is being transmitted. Two types of passive attacks are the release of message contents and traffic analysis.

(a) Passive attacks

Alice

(b) Active attacks

Bob

Darth

Bob

Darth

Alice

Internet orother communications facility

Internet orother communications facility

1 2

3

Figure 1.1 Security Attacks


The release of message contents is easily understood. A telephone conver-sation, an electronic mail message, and a transferred file may contain sensitive or confidential information. We would like to prevent an opponent from learning the contents of these transmissions.

A second type of passive attack, traffic analysis, is subtler. Suppose that we had a way of masking the contents of messages or other information traffic so that opponents, even if they captured the message, could not extract the information from the message. The common technique for masking contents is encryption. If we had encryption protection in place, an opponent might still be able to observe the pattern of these messages. The opponent could determine the location and identity of communicating hosts and could observe the frequency and length of messages being exchanged. This information might be useful in guessing the nature of the communication that was taking place.

Passive attacks are very difficult to detect, because they do not involve any alteration of the data. Typically, the message traffic is sent and received in an appar-ently normal fashion, and neither the sender nor receiver is aware that a third party has read the messages or observed the traffic pattern. However, it is feasible to pre-vent the success of these attacks, usually by means of encryption. Thus, the emphasis in dealing with passive attacks is on prevention rather than detection.

Active Attacks

Active attacks (Figure 1.1b) involve some modification of the data stream or the creation of a false stream and can be subdivided into four categories: masquerade, replay, modification of messages, and denial of service.

A masquerade takes place when one entity pretends to be a different entity (path 2 of Figure 1.1b is active). A masquerade attack usually includes one of the other forms of active attack. For example, authentication sequences can be captured and replayed after a valid authentication sequence has taken place, thus enabling an authorized entity with few privileges to obtain extra privileges by impersonating an entity that has those privileges.

Replay involves the passive capture of a data unit and its subsequent retrans-mission to produce an unauthorized effect (paths 1, 2, and 3 active).

Modification of messages simply means that some portion of a legitimate message is altered, or that messages are delayed or reordered, to produce an unauthorized effect (paths 1 and 2 active). For example, a message meaning “Allow John Smith to read confidential file accounts” is modified to mean “Allow Fred Brown to read confidential file accounts.”

The denial of service prevents or inhibits the normal use or management of communications facilities (path 3 active). This attack may have a specific target; for example, an entity may suppress all messages directed to a particular destination (e.g., the security audit service). Another form of service denial is the disruption of an entire network, either by disabling the network or by overloading it with messages so as to degrade performance.

Active attacks present the opposite characteristics of passive attacks. Whereas passive attacks are difficult to detect, measures are available to prevent their suc-cess. On the other hand, it is quite difficult to prevent active attacks absolutely

1.4 / SeCurity ServiCeS 17

because of the wide variety of potential physical, software, and network vulner-abilities. Instead, the goal is to detect active attacks and to recover from any dis-ruption or delays caused by them. If the detection has a deterrent effect, it may also contribute to prevention.

1.4 security services

X.800 defines a security service as a service that is provided by a protocol layer of communicating open systems and that ensures adequate security of the systems or of data transfers. Perhaps a clearer definition is found in RFC 4949, which provides the following definition: a processing or communication service that is provided by a system to give a specific kind of protection to system resources; security services implement security policies and are implemented by security mechanisms.

X.800 divides these services into five categories and fourteen specific services (Table 1.2). We look at each category in turn.5

Authentication

The authentication service is concerned with assuring that a communication is authentic. In the case of a single message, such as a warning or alarm signal, the function of the authentication service is to assure the recipient that the message is from the source that it claims to be from. In the case of an ongoing interaction, such as the connection of a terminal to a host, two aspects are involved. First, at the time of connection initiation, the service assures that the two entities are authentic, that is, that each is the entity that it claims to be. Second, the service must assure that the connection is not interfered with in such a way that a third party can masquerade as one of the two legitimate parties for the purposes of unauthorized transmission or reception.

Two specific authentication services are defined in X.800:

• Peer entity authentication: Provides for the corroboration of the identity of a peer entity in an association. Two entities are considered peers if they implement to same protocol in different systems; for example two TCP mod-ules in two communicating systems. Peer entity authentication is provided for use at the establishment of, or at times during the data transfer phase of, a connection. It attempts to provide confidence that an entity is not performing either a masquerade or an unauthorized replay of a previous connection.

• Data origin authentication: Provides for the corroboration of the source of a data unit. It does not provide protection against the duplication or modification of data units. This type of service supports applications like electronic mail, where there are no prior interactions between the communicating entities.

5There is no universal agreement about many of the terms used in the security literature. For example, the term integrity is sometimes used to refer to all aspects of information security. The term authentication is sometimes used to refer both to verification of identity and to the various functions listed under integrity in this chapter. Our usage here agrees with both X.800 and RFC 4949.


Access Control

In the context of network security, access control is the ability to limit and control the access to host systems and applications via communications links. To achieve this, each entity trying to gain access must first be identified, or authenticated, so that access rights can be tailored to the individual.

Table 1.2 Security Services (X.800)

AUTHENTICATION

The assurance that the communicating entity is the one that it claims to be.

Peer Entity AuthenticationUsed in association with a logical connection to provide confidence in the identity of the entities connected.

Data-Origin AuthenticationIn a connectionless transfer, provides assurance that the source of received data is as claimed.

ACCESS CONTROL

The prevention of unauthorized use of a resource (i.e., this service controls who can have access to a resource, under what conditions access can occur, and what those accessing the resource are allowed to do).

DATA CONFIDENTIALITY

The protection of data from unauthorized disclosure.

Connection ConfidentialityThe protection of all user data on a connection.

Connectionless ConfidentialityThe protection of all user data in a single data block

Selective-Field ConfidentialityThe confidentiality of selected fields within the user data on a connection or in a single data block.

Traffic-Flow ConfidentialityThe protection of the information that might be derived from observation of traffic flows.

DATA INTEGRITY

The assurance that data received are exactly as sent by an authorized entity (i.e., contain no modification, insertion, deletion, or replay).

Connection Integrity with RecoveryProvides for the integrity of all user data on a connection and detects any modification, insertion, deletion, or replay of any data within an entire data sequence, with recovery attempted.

Connection Integrity without RecoveryAs above, but provides only detection without recovery.

Selective-Field Connection IntegrityProvides for the integrity of selected fields within the user data of a data block transferred over a connec-tion and takes the form of determination of whether the selected fields have been modified, inserted, deleted, or replayed.

Connectionless IntegrityProvides for the integrity of a single connectionless data block and may take the form of detection of data modification. Additionally, a limited form of replay detection may be provided.

Selective-Field Connectionless IntegrityProvides for the integrity of selected fields within a single connectionless data block; takes the form of determination of whether the selected fields have been modified.

NONREPUDIATION

Provides protection against denial by one of the entities involved in a communication of having participated in all or part of the communication.

Nonrepudiation, OriginProof that the message was sent by the specified party.

Nonrepudiation, DestinationProof that the message was received by the specified party.

1.4 / SeCurity ServiCeS 19

Data Confidentiality

Confidentiality is the protection of transmitted data from passive attacks. With respect to the content of a data transmission, several levels of protection can be identified. The broadest service protects all user data transmitted between two users over a period of time. For example, when a TCP connection is set up between two systems, this broad protection prevents the release of any user data transmit-ted over the TCP connection. Narrower forms of this service can also be defined, including the protection of a single message or even specific fields within a message. These refinements are less useful than the broad approach and may even be more complex and expensive to implement.

The other aspect of confidentiality is the protection of traffic flow from analysis. This requires that an attacker not be able to observe the source and destination, fre-quency, length, or other characteristics of the traffic on a communications facility.

Data Integrity

As with confidentiality, integrity can apply to a stream of messages, a single mes-sage, or selected fields within a message. Again, the most useful and straightforward approach is total stream protection.

A connection-oriented integrity service, one that deals with a stream of mes-sages, assures that messages are received as sent with no duplication, insertion, modification, reordering, or replays. The destruction of data is also covered under this service. Thus, the connection-oriented integrity service addresses both message stream modification and denial of service. On the other hand, a connectionless in-tegrity service, one that deals with individual messages without regard to any larger context, generally provides protection against message modification only.

We can make a distinction between service with and without recovery. Because the integrity service relates to active attacks, we are concerned with detec-tion rather than prevention. If a violation of integrity is detected, then the service may simply report this violation, and some other portion of software or human intervention is required to recover from the violation. Alternatively, there are mechanisms available to recover from the loss of integrity of data, as we will review subsequently. The incorporation of automated recovery mechanisms is, in general, the more attractive alternative.

Nonrepudiation

Nonrepudiation prevents either sender or receiver from denying a transmitted mes-sage. Thus, when a message is sent, the receiver can prove that the alleged sender in fact sent the message. Similarly, when a message is received, the sender can prove that the alleged receiver in fact received the message.

Availability Service

Both X.800 and RFC 4949 define availability to be the property of a system or a system resource being accessible and usable upon demand by an authorized system entity, according to performance specifications for the system (i.e., a system is avail-able if it provides services according to the system design whenever users request


them). A variety of attacks can result in the loss of or reduction in availability. Some of these attacks are amenable to automated countermeasures, such as authentica-tion and encryption, whereas others require some sort of physical action to prevent or recover from loss of availability of elements of a distributed system.

X.800 treats availability as a property to be associated with various security services. However, it makes sense to call out specifically an availability service. An availability service is one that protects a system to ensure its availability. This ser-vice addresses the security concerns raised by denial-of-service attacks. It depends on proper management and control of system resources and thus depends on access control service and other security services.

1.5 security mechanisms

Table 1.3 lists the security mechanisms defined in X.800. The mechanisms are divided into those that are implemented in a specific protocol layer, such as TCP or an application-layer protocol, and those that are not specific to any particu-lar protocol layer or security service. These mechanisms will be covered in the appropriate places in the book. So we do not elaborate now, except to comment on the definition of encipherment. X.800 distinguishes between reversible enci-pherment mechanisms and irreversible encipherment mechanisms. A reversible

Table 1.3 Security Mechanisms (X.800)

SPECIFIC SECURITY MECHANISMS

May be incorporated into the appropriate protocol layer in order to provide some of the OSI security services.

EnciphermentThe use of mathematical algorithms to transform data into a form that is not readily intelligible. The transformation and subsequent recovery of the data depend on an algorithm and zero or more encryption keys.

Digital SignatureData appended to, or a cryptographic transformation of, a data unit that allows a recipient of the data unit to prove the source and integrity of the data unit and protect against forgery (e.g., by the recipient).

Access ControlA variety of mechanisms that enforce access rights to resources.

Data IntegrityA variety of mechanisms used to assure the integrity of a data unit or stream of data units.

PERVASIVE SECURITY MECHANISMS

Mechanisms that are not specific to any particular OSI security service or protocol layer.

Trusted FunctionalityThat which is perceived to be correct with respect to some criteria (e.g., as established by a security policy).

Security LabelThe marking bound to a resource (which may be a data unit) that names or designates the security attri-butes of that resource.

Event DetectionDetection of security-relevant events.

Security Audit TrailData collected and potentially used to facilitate a security audit, which is an independent review and examination of system records and activities.

Security RecoveryDeals with requests from mechanisms, such as event handling and management functions, and takes recovery actions.

1.5 / SeCurity meChaniSmS 21

encipherment mechanism is simply an encryption algorithm that allows data to be encrypted and subsequently decrypted. Irreversible encipherment mechanisms include hash algorithms and message authentication codes, which are used in digi-tal signature and message authentication applications.

Table 1.4, based on one in X.800, indicates the relationship between security services and security mechanisms.

SPECIFIC SECURITY MECHANISMS

Authentication ExchangeA mechanism intended to ensure the identity of an entity by means of information exchange.

Traffic PaddingThe insertion of bits into gaps in a data stream to frustrate traffic analysis attempts.

Routing ControlEnables selection of particular physically secure routes for certain data and allows routing changes, especially when a breach of security is suspected.

NotarizationThe use of a trusted third party to assure certain properties of a data exchange.

Table 1.3 Continued

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Peer entity authentication

SERVICE

MECHANISM

Enchip

herm

ent

Digital

sign

ature

Acces

s con

trol

Data in

tegrit

y

Authen

ticati

on ex

chan

ge

Traffic

padd

ing

Routin

g con

trol

Notariz

ation

Data origin authentication

Access control

Confidentiality

Traffic flow confidentiality

Data integrity

Nonrepudiation

Availability

Table 1.4 Relationship Between Security Services and Mechanisms


InformationchannelSecurity-related

transformation

Sender

Secretinformation

Mes

sage

Mes

sage

Secu

rem

essa

ge

Secu

rem

essa

ge

Recipient

Opponent

Trusted third party(e.g., arbiter, distributerof secret information)

Security-relatedtransformation

Secretinformation

Figure 1.2 Model for Network Security

1.6 a mOdeL fOr netwOrk security

A model for much of what we will be discussing is captured, in very general terms, in Figure 1.2. A message is to be transferred from one party to another across some sort of Internet service. The two parties, who are the principals in this transaction, must cooperate for the exchange to take place. A logical information channel is established by defining a route through the Internet from source to destination and by the coop-erative use of communication protocols (e.g., TCP/IP) by the two principals.

Security aspects come into play when it is necessary or desirable to protect the in-formation transmission from an opponent who may present a threat to confidentiality, authenticity, and so on. All the techniques for providing security have two components:

• A security-related transformation on the information to be sent. Examples include the encryption of the message, which scrambles the message so that it is unreadable by the opponent, and the addition of a code based on the con-tents of the message, which can be used to verify the identity of the sender.

• Some secret information shared by the two principals and, it is hoped, unknown to the opponent. An example is an encryption key used in conjunction with the transformation to scramble the message before transmission and unscramble it on reception.6

A trusted third party may be needed to achieve secure transmission. For example, a third party may be responsible for distributing the secret information

6Part Two discusses a form of encryption, known as a symmetric encryption, in which only one of the two principals needs to have the secret information.

1.6 / a mOdel fOr netwOrk SeCurity 23

to the two principals while keeping it from any opponent. Or a third party may be needed to arbitrate disputes between the two principals concerning the authentic-ity of a message transmission.

This general model shows that there are four basic tasks in designing a particu-lar security service:

1. Design an algorithm for performing the security-related transformation. The algorithm should be such that an opponent cannot defeat its purpose.

2. Generate the secret information to be used with the algorithm.

3. Develop methods for the distribution and sharing of the secret information.

4. Specify a protocol to be used by the two principals that makes use of the security algorithm and the secret information to achieve a particular security service.

Parts One through Five of this book concentrate on the types of security mech-anisms and services that fit into the model shown in Figure 1.2. However, there are other security-related situations of interest that do not neatly fit this model but are considered in this book. A general model of these other situations is illustrated in Figure 1.3, which reflects a concern for protecting an information system from un-wanted access. Most readers are familiar with the concerns caused by the existence of hackers, who attempt to penetrate systems that can be accessed over a network. The hacker can be someone who, with no malign intent, simply gets satisfaction from breaking and entering a computer system. The intruder can be a disgruntled employee who wishes to do damage or a criminal who seeks to exploit computer assets for financial gain (e.g., obtaining credit card numbers or performing illegal money transfers).

Another type of unwanted access is the placement in a computer system of logic that exploits vulnerabilities in the system and that can affect application programs as well as utility programs, such as editors and compilers. Programs can present two kinds of threats:

• Information access threats: Intercept or modify data on behalf of users who should not have access to that data.

• Service threats: Exploit service flaws in computers to inhibit use by legitimate users.

Computing resources (processor, memory, I/O)

Data

Processes

Software

Internal security controls

Information system

Gatekeeperfunction

Opponent—human (e.g., hacker)

—software (e.g., virus, worm)

Access channel

Figure 1.3 Network Access Security Model


Viruses and worms are two examples of software attacks. Such attacks can be introduced into a system by means of a disk that contains the unwanted logic con-cealed in otherwise useful software. They can also be inserted into a system across a network; this latter mechanism is of more concern in network security.

The security mechanisms needed to cope with unwanted access fall into two broad categories (see Figure 1.3). The first category might be termed a gate-keeper function. It includes password-based login procedures that are designed to deny access to all but authorized users and screening logic that is designed to detect and reject worms, viruses, and other similar attacks. Once either an unwanted user or unwanted software gains access, the second line of defense consists of a variety of internal controls that monitor activity and analyze stored information in an attempt to detect the presence of unwanted intruders. These issues are explored in Part Six.

1.7 recOmmended reading

[STAL12] provides a broad introduction to both computer and network security. [SCHN00] is valuable reading for any practitioner in the field of computer or network security: It discusses the limitations of technology, and cryptography in particular, in providing security and the need to consider the hardware, the software implementation, the networks, and the people involved in providing and attacking security.

It is useful to read some of the classic tutorial papers on computer security; these provide a historical perspective from which to appreciate current work and thinking.7 The papers to read are [WARE79], [BROW72], [SALT75], [SHAN77], and [SUMM84]. Two more recent, short treatments of computer security are [ANDR04] and [LAMP04]. [NIST95] is an exhaustive (290 pages) treatment of the subject. Another good treatment is [NRC91]. Also useful is [FRAS97].

7These classic papers are available in the Premium Content Web site for this book.

ANDR04 Andrews, M., and Whittaker, J. “Computer Security.” IEEE Security and Privacy, September/October 2004.

BROW72 Browne, P. “Computer Security—A Survey.” ACM SIGMIS Database, Fall 1972.

FRAS97 Fraser, B. Site Security Handbook. RFC 2196, September 1997.LAMP04 Lampson, B. “Computer Security in the Real World,” Computer, June 2004.NIST95 National Institute of Standards and Technology. An Introduction to Computer

Security: The NIST Handbook. Special Publication 800–12, October 1995.NRC91 National Research Council. Computers at Risk: Safe Computing in the

Information Age. Washington, D.C.: National Academy Press, 1991.SALT75 Saltzer, J., and Schroeder, M. “The Protection of Information in Computer

Systems.” Proceedings of the IEEE, September 1975.SCHN00 Schneier, B. Secrets and Lies: Digital Security in a Networked World.

New York: Wiley, 2000.

1.8 / key termS, review QueStiOnS, and prOblemS 25

SHAN77 Shanker, K. “The Total Computer Security Problem: An Overview.” Computer, June 1977.

STAL12 Stallings, W., and Brown, L. Computer Security. Upper Saddle River, NJ: Prentice Hall, 2012.

SUMM84 Summers, R. “An Overview of Computer Security.” IBM Systems Journal, Vol. 23, No. 4, 1984.

WARE79 Ware, W., ed. Security Controls for Computer Systems. RAND Report 609–1. October 1979.

1.8 key terms, review QuestiOns, and prObLems

Key Terms

access controlactive attackauthenticationauthenticityavailabilitydata confidentialitydata integrity

denial of serviceencryptionintegrityintrudermasqueradenonrepudiationOSI security architecture

passive attackreplaysecurity attackssecurity mechanismssecurity servicestraffic analysis

Review Questions 1.1 What is the OSI security architecture? 1.2 What is the difference between passive and active security threats? 1.3 List and briefly define categories of passive and active security attacks. 1.4 List and briefly define categories of security services. 1.5 List and briefly define categories of security mechanisms.

Problems 1.1 Consider an automated teller machine (ATM) in which users provide a personal

identification number (PIN) and a card for account access. Give examples of confi-dentiality, integrity, and availability requirements associated with the system and, in each case, indicate the degree of importance of the requirement.

1.2 Repeat Problem 1.1 for a telephone switching system that routes calls through a switching network based on the telephone number requested by the caller.

1.3 Consider a desktop publishing system used to produce documents for various organizations.

a. Give an example of a type of publication for which confidentiality of the stored data is the most important requirement.

b. Give an example of a type of publication in which data integrity is the most impor-tant requirement.

c. Give an example in which system availability is the most important requirement.


1.4 For each of the following assets, assign a low, moderate, or high impact level for the loss of confidentiality, availability, and integrity, respectively. Justify your answers.

a. An organization managing public information on its Web server. b. A law enforcement organization managing extremely sensitive investigative

information. c. A financial organization managing routine administrative information (not

privacy-related information). d. An information system used for large acquisitions in a contracting organization

contains both sensitive, pre-solicitation phase contract information and routine administrative information. Assess the impact for the two data sets separately and the information system as a whole.

e. A power plant contains a SCADA (supervisory control and data acquisition) system controlling the distribution of electric power for a large military installa-tion. The SCADA system contains both real-time sensor data and routine admin-istrative information. Assess the impact for the two data sets separately and the information system as a whole.

1.5 Draw a matrix similar to Table 1.4 that shows the relationship between security services and attacks.

1.6 Draw a matrix similar to Table 1.4 that shows the relationship between security mechanisms and attacks.

1.7 Read all of the classic papers cited in Section 1.7. Compose a 500–1000 word paper (or 8–12 slide PowerPoint presentation) that summarizes the key concepts that emerge from these papers, emphasizing concepts that are common to most or all of the papers.

27

Part 1: Symmetric ciPherS

Chapter

ClassiCal EnCryption tEChniquEs

2.1 Symmetric Cipher Model

CryptographyCryptanalysis and Brute-Force Attack

2.2 Substitution Techniques

Caesar CipherMonoalphabetic CiphersPlayfair CipherHill CipherPolyalphabetic CiphersOne-Time Pad

2.3 Transposition Techniques

2.4 Rotor Machines

2.5 Steganography



28 Chapter 2 / ClassiCal enCryption teChniques

“I am fairly familiar with all the forms of secret writings, and am myself the author of a trifling monograph upon the subject, in which I analyze one hundred and sixty separate ciphers,” said Holmes.

—The Adventure of the Dancing Men, Sir Arthur Conan Doyle

Learning Objectives


u Present an overview of the main concepts of symmetric cryptography.

u Explain the difference between cryptanalysis and brute-force attack.

u Understand the operation of a monoalphabetic substitution cipher.

u Understand the operation of a polyalphabetic cipher.

u Present an overview of the Hill cipher.

u Describe the operation of a rotor machine.

Symmetric encryption, also referred to as conventional encryption or single-key encryption, was the only type of encryption in use prior to the development of public-key encryption in the 1970s. It remains by far the most widely used of the two types of encryption. Part One examines a number of symmetric ciphers. In this chapter, we begin with a look at a general model for the symmetric encryption process; this will enable us to understand the context within which the algorithms are used. Next, we examine a variety of algorithms in use before the computer era. Finally, we look briefly at a different approach known as steganography. Chapters 3 and 5 introduce the two most widely used symmetric cipher: DES and AES.

Before beginning, we define some terms. An original message is known as the plaintext, while the coded message is called the ciphertext. The process of convert-ing from plaintext to ciphertext is known as enciphering or encryption; restoring the plaintext from the ciphertext is deciphering or decryption. The many schemes used for encryption constitute the area of study known as cryptography. Such a scheme is known as a cryptographic system or a cipher. Techniques used for decipher-ing a message without any knowledge of the enciphering details fall into the area of cryptanalysis. Cryptanalysis is what the layperson calls “breaking the code.” The areas of cryptography and cryptanalysis together are called cryptology.

2.1 symmetric cipher mOdeL

A symmetric encryption scheme has five ingredients (Figure 2.1):

• Plaintext: This is the original intelligible message or data that is fed into the algorithm as input.

2.1 / symmetriC Cipher model 29

• Encryption algorithm: The encryption algorithm performs various substitu-tions and transformations on the plaintext.

• Secret key: The secret key is also input to the encryption algorithm. The key is a value independent of the plaintext and of the algorithm. The algorithm will produce a different output depending on the specific key being used at the time. The exact substitutions and transformations performed by the algorithm depend on the key.

• Ciphertext: This is the scrambled message produced as output. It depends on the plaintext and the secret key. For a given message, two different keys will produce two different ciphertexts. The ciphertext is an apparently random stream of data and, as it stands, is unintelligible.

• Decryption algorithm: This is essentially the encryption algorithm run in reverse. It takes the ciphertext and the secret key and produces the original plaintext.

There are two requirements for secure use of conventional encryption:

1. We need a strong encryption algorithm. At a minimum, we would like the algorithm to be such that an opponent who knows the algorithm and has access to one or more ciphertexts would be unable to decipher the ciphertext or figure out the key. This requirement is usually stated in a stronger form: The opponent should be unable to decrypt ciphertext or discover the key even if he or she is in possession of a number of ciphertexts together with the plain-text that produced each ciphertext.

2. Sender and receiver must have obtained copies of the secret key in a secure fashion and must keep the key secure. If someone can discover the key and knows the algorithm, all communication using this key is readable.

We assume that it is impractical to decrypt a message on the basis of the ciphertext plus knowledge of the encryption/decryption algorithm. In other words, we do not need to keep the algorithm secret; we need to keep only the key secret. This feature of symmetric encryption is what makes it feasible for widespread use. The fact that the algorithm need not be kept secret means that manufacturers can and have

Plaintextinput

Y = E(K, X) X = D(K, Y)

X

KK

Transmittedciphertext

Plaintextoutput

Secret key shared bysender and recipient

Secret key shared bysender and recipient

Encryption algorithm(e.g., AES)

Decryption algorithm(reverse of encryption

algorithm)

Figure 2.1 Simplified Model of Symmetric Encryption


developed low-cost chip implementations of data encryption algorithms. These chips are widely available and incorporated into a number of products. With the use of sym-metric encryption, the principal security problem is maintaining the secrecy of the key.

Let us take a closer look at the essential elements of a symmetric encryption scheme, using Figure 2.2. A source produces a message in plaintext, X = [X1, X2, c, XM]. The M elements of X are letters in some finite alphabet. Traditionally, the alphabet usually consisted of the 26 capital letters. Nowadays, the binary alphabet {0, 1} is typically used. For encryption, a key of the form K = [K1, K2, c, KJ] is generated. If the key is generated at the message source, then it must also be provided to the destination by means of some secure channel. Alternatively, a third party could generate the key and securely deliver it to both source and destination.

With the message X and the encryption key K as input, the encryption algo-rithm forms the ciphertext Y = [Y1, Y2, c, YN]. We can write this as

Y = E(K, X)

This notation indicates that Y is produced by using encryption algorithm E as a function of the plaintext X, with the specific function determined by the value of the key K.

The intended receiver, in possession of the key, is able to invert the transformation:

X = D(K, Y)

An opponent, observing Y but not having access to K or X , may attempt to recover X or K or both X and K. It is assumed that the opponent knows the

Messagesource

Cryptanalyst

Keysource

DestinationX X

X̂

K̂

Y = E(K, X)

Secure channel

K

Encryptionalgorithm

Decryptionalgorithm

Figure 2.2 Model of Symmetric Cryptosystem


encryption (E) and decryption (D) algorithms. If the opponent is interested in only this particular message, then the focus of the effort is to recover X by generating a plaintext estimate Xn . Often, however, the opponent is interested in being able to read future messages as well, in which case an attempt is made to recover K by generating an estimate Kn .

Cryptography

Cryptographic systems are characterized along three independent dimensions:

1. The type of operations used for transforming plaintext to ciphertext. All encryption algorithms are based on two general principles: substitution, in which each element in the plaintext (bit, letter, group of bits or letters) is mapped into another element, and transposition, in which elements in the plaintext are rearranged. The fundamental requirement is that no information be lost (i.e., that all operations are reversible). Most systems, referred to as product systems, involve multiple stages of substitutions and transpositions.

2. The number of keys used. If both sender and receiver use the same key, the system is referred to as symmetric, single-key, secret-key, or conventional encryption. If the sender and receiver use different keys, the system is referred to as asymmetric, two-key, or public-key encryption.

3. The way in which the plaintext is processed. A block cipher processes the input one block of elements at a time, producing an output block for each input block. A stream cipher processes the input elements continuously, producing output one element at a time, as it goes along.

Cryptanalysis and Brute-Force Attack

Typically, the objective of attacking an encryption system is to recover the key in use rather than simply to recover the plaintext of a single ciphertext. There are two general approaches to attacking a conventional encryption scheme:

• Cryptanalysis: Cryptanalytic attacks rely on the nature of the algorithm plus perhaps some knowledge of the general characteristics of the plaintext or even some sample plaintext–ciphertext pairs. This type of attack exploits the characteristics of the algorithm to attempt to deduce a specific plaintext or to deduce the key being used.

• Brute-force attack: The attacker tries every possible key on a piece of cipher-text until an intelligible translation into plaintext is obtained. On average, half of all possible keys must be tried to achieve success.

If either type of attack succeeds in deducing the key, the effect is catastrophic: All future and past messages encrypted with that key are compromised.

We first consider cryptanalysis and then discuss brute-force attacks.Table 2.1 summarizes the various types of cryptanalytic attacks based on the

amount of information known to the cryptanalyst. The most difficult problem is presented when all that is available is the ciphertext only. In some cases, not even the encryption algorithm is known, but in general, we can assume that the oppo-nent does know the algorithm used for encryption. One possible attack under these


circumstances is the brute-force approach of trying all possible keys. If the key space is very large, this becomes impractical. Thus, the opponent must rely on an analysis of the ciphertext itself, generally applying various statistical tests to it. To use this approach, the opponent must have some general idea of the type of plaintext that is concealed, such as English or French text, an EXE file, a Java source listing, an accounting file, and so on.

The ciphertext-only attack is the easiest to defend against because the opponent has the least amount of information to work with. In many cases, however, the analyst has more information. The analyst may be able to capture one or more plaintext messages as well as their encryptions. Or the analyst may know that certain plaintext patterns will appear in a message. For example, a file that is encoded in the Postscript format always begins with the same pattern, or there may be a standardized header or banner to an electronic funds transfer message, and so on. All these are examples of known plaintext. With this knowledge, the analyst may be able to deduce the key on the basis of the way in which the known plaintext is transformed.

Closely related to the known-plaintext attack is what might be referred to as a probable-word attack. If the opponent is working with the encryption of some gen-eral prose message, he or she may have little knowledge of what is in the message. However, if the opponent is after some very specific information, then parts of the message may be known. For example, if an entire accounting file is being transmit-ted, the opponent may know the placement of certain key words in the header of the file. As another example, the source code for a program developed by Corporation X might include a copyright statement in some standardized position.

Table 2.1 Types of Attacks on Encrypted Messages

Type of Attack Known to Cryptanalyst

Ciphertext Only •Encryptionalgorithm

•Ciphertext

Known Plaintext •Encryptionalgorithm

•Ciphertext

•Oneormoreplaintext–ciphertextpairsformedwiththesecretkey

Chosen Plaintext •Encryptionalgorithm

•Ciphertext

•Plaintextmessagechosenbycryptanalyst,togetherwithitscorresponding ciphertext generated with the secret key

Chosen Ciphertext •Encryptionalgorithm

•Ciphertext

•Ciphertextchosenbycryptanalyst,togetherwithitscorrespondingdecryptedplaintext generated with the secret key

Chosen Text •Encryptionalgorithm

•Ciphertext

•Plaintextmessagechosenbycryptanalyst,togetherwithitscorresponding ciphertext generated with the secret key

•Ciphertextchosenbycryptanalyst,togetherwithitscorrespondingdecryptedplaintext generated with the secret key


If the analyst is able somehow to get the source system to insert into the sys-tem a message chosen by the analyst, then a chosen-plaintext attack is possible. An example of this strategy is differential cryptanalysis, explored in Chapter 3. In general, if the analyst is able to choose the messages to encrypt, the analyst may deliberately pick patterns that can be expected to reveal the structure of the key.

Table 2.1 lists two other types of attack: chosen ciphertext and chosen text. These are less commonly employed as cryptanalytic techniques but are nevertheless possible avenues of attack.

Only relatively weak algorithms fail to withstand a ciphertext-only attack. Generally, an encryption algorithm is designed to withstand a known-plaintext attack.

Two more definitions are worthy of note. An encryption scheme is uncondi-tionally secure if the ciphertext generated by the scheme does not contain enough information to determine uniquely the corresponding plaintext, no matter how much ciphertext is available. That is, no matter how much time an opponent has, it is impossible for him or her to decrypt the ciphertext simply because the required information is not there. With the exception of a scheme known as the one-time pad (described later in this chapter), there is no encryption algorithm that is uncondi-tionally secure. Therefore, all that the users of an encryption algorithm can strive for is an algorithm that meets one or both of the following criteria:

• The cost of breaking the cipher exceeds the value of the encrypted information. • The time required to break the cipher exceeds the useful lifetime of the

information.

An encryption scheme is said to be computationally secure if either of the foregoing two criteria are met. Unfortunately, it is very difficult to estimate the amount of effort required to cryptanalyze ciphertext successfully.

All forms of cryptanalysis for symmetric encryption schemes are designed to exploit the fact that traces of structure or pattern in the plaintext may survive encryption and be discernible in the ciphertext. This will become clear as we exam-ine various symmetric encryption schemes in this chapter. We will see in Part Two that cryptanalysis for public-key schemes proceeds from a fundamentally different premise, namely, that the mathematical properties of the pair of keys may make it possible for one of the two keys to be deduced from the other.

A brute-force attack involves trying every possible key until an intelligible translation of the ciphertext into plaintext is obtained. On average, half of all pos-sible keys must be tried to achieve success. That is, if there are X different keys, on average an attacker would discover the actual key after X>2 tries. It is important to note that there is more to a brute-force attack than simply running through all pos-sible keys. Unless known plaintext is provided, the analyst must be able to recognize plaintext as plaintext. If the message is just plain text in English, then the result pops out easily, although the task of recognizing English would have to be automated. If the text message has been compressed before encryption, then recognition is more difficult. And if the message is some more general type of data, such as a numeri-cal file, and this has been compressed, the problem becomes even more difficult to automate. Thus, to supplement the brute-force approach, some degree of knowl-edge about the expected plaintext is needed, and some means of automatically distinguishing plaintext from garble is also needed.


2.2 substitutiOn techniques

In this section and the next, we examine a sampling of what might be called classi-cal encryption techniques. A study of these techniques enables us to illustrate the basic approaches to symmetric encryption used today and the types of cryptanalytic attacks that must be anticipated.

The two basic building blocks of all encryption techniques are substitution and transposition. We examine these in the next two sections. Finally, we discuss a system that combines both substitution and transposition.

A substitution technique is one in which the letters of plaintext are replaced by other letters or by numbers or symbols.1 If the plaintext is viewed as a sequence of bits, then substitution involves replacing plaintext bit patterns with ciphertext bit patterns.

Caesar Cipher

The earliest known, and the simplest, use of a substitution cipher was by Julius Caesar. The Caesar cipher involves replacing each letter of the alphabet with the letter standing three places further down the alphabet. For example,

plain: meet me after the toga party

cipher: PHHW PH DIWHU WKH WRJD SDUWB

Note that the alphabet is wrapped around, so that the letter following Z is A. We can define the transformation by listing all possibilities, as follows:

plain: a b c d e f g h i j k l m n o p q r s t u v w x y z

cipher: D e f g H I J K l m n o P q R S T U v W x y z a B c

Let us assign a numerical equivalent to each letter:

a b c d e f g h i j k l m0 1 2 3 4 5 6 7 8 9 10 11 12

n o p q r s t u v w x y z

13 14 15 16 17 18 19 20 21 22 23 24 25

Then the algorithm can be expressed as follows. For each plaintext letter p, substi-tute the ciphertext letter C:2

C = E(3, p) = (p + 3) mod 26

1When letters are involved, the following conventions are used in this book. Plaintext is always in lowercase; ciphertext is in uppercase; key values are in italicized lowercase.2We define a mod n to be the remainder when a is divided by n. For example, 11 mod 7 = 4. See Chapter 4 for a further discussion of modular arithmetic.

2.2 / substitution teChniques 35

A shift may be of any amount, so that the general Caesar algorithm is

C = E(k, p) = (p + k) mod 26 (2.1)

where k takes on a value in the range 1 to 25. The decryption algorithm is simply

p = D(k, C) = (C - k) mod 26 (2.2)

If it is known that a given ciphertext is a Caesar cipher, then a brute-force cryptanalysis is easily performed: simply try all the 25 possible keys. Figure 2.3 shows the results of applying this strategy to the example ciphertext. In this case, the plaintext leaps out as occupying the third line.

Three important characteristics of this problem enabled us to use a brute-force cryptanalysis:

1. The encryption and decryption algorithms are known.

2. There are only 25 keys to try.

3. The language of the plaintext is known and easily recognizable.

PHHW PH DIWHU WKH WRJD SDUWBKEY

1 oggv og chvgt vjg vqic rctva

2 nffu nf bgufs uif uphb qbsuz

3 meet me after the toga party

4 ldds ld zesdq sgd snfz ozqsx

5 kccr kc ydrcp rfc rmey nyprw

6 jbbq jb xcqbo qeb qldx mxoqv

7 iaap ia wbpan pda pkcw lwnpu

8 hzzo hz vaozm ocz ojbv kvmot

9 gyyn gy uznyl nby niau julns

10 fxxm fx tymxk max mhzt itkmr

11 ewwl ew sxlwj lzw lgys hsjlq

12 dvvk dv rwkvi kyv kfxr grikp

13 cuuj cu qvjuh jxu jewq fqhjo

14 btti bt puitg iwt idvp epgin

15 assh as othsf hvs hcuo dofhm

16 zrrg zr nsgre gur gbtn cnegl

17 yqqf yq mrfqd ftq fasm bmdfk

18 xppe xp lqepc esp ezrl alcej

19 wood wo kpdob dro dyqk zkbdi

20 vnnc vn jocna cqn cxpj yjach

21 ummb um inbmz bpm bwoi xizbg

22 tlla tl hmaly aol avnh whyaf

23 skkz sk glzkx znk zumg vgxze

24 rjjy rj fkyjw ymj ytlf ufwyd

25 qiix qi ejxiv xli xske tevxc

Figure 2.3 Brute-Force Cryptanalysis of Caesar Cipher


In most networking situations, we can assume that the algorithms are known. What generally makes brute-force cryptanalysis impractical is the use of an algo-rithm that employs a large number of keys. For example, the triple DES algorithm, examined in Chapter 6, makes use of a 168-bit key, giving a key space of 2168 or greater than 3.7 * 1050 possible keys.

The third characteristic is also significant. If the language of the plaintext is unknown, then plaintext output may not be recognizable. Furthermore, the input may be abbreviated or compressed in some fashion, again making recogni-tion difficult. For example, Figure 2.4 shows a portion of a text file compressed using an algorithm called ZIP. If this file is then encrypted with a simple sub-stitution cipher (expanded to include more than just 26 alphabetic characters), then the plaintext may not be recognized when it is uncovered in the brute-force cryptanalysis.

Monoalphabetic Ciphers

With only 25 possible keys, the Caesar cipher is far from secure. A dramatic increase in the key space can be achieved by allowing an arbitrary substitution. Before pro-ceeding, we define the term permutation. A permutation of a finite set of elements S is an ordered sequence of all the elements of S, with each element appearing exactly once. For example, if S = {a, b, c}, there are six permutations of S:

abc, acb, bac, bca, cab, cba

In general, there are n! permutations of a set of n elements, because the first element can be chosen in one of n ways, the second in n - 1 ways, the third in n - 2 ways, and so on.

Recall the assignment for the Caesar cipher:

plain: a b c d e f g h i j k l m n o p q r s t u v w x y z

cipher: D e f g H I J K l m n o P q R S T U v W x y z a B c

If, instead, the “cipher” line can be any permutation of the 26 alphabetic characters, then there are 26! or greater than 4 * 1026 possible keys. This is 10 orders of mag-nitude greater than the key space for DES and would seem to eliminate brute-force techniques for cryptanalysis. Such an approach is referred to as a monoalphabetic substitution cipher, because a single cipher alphabet (mapping from plain alphabet to cipher alphabet) is used per message.

Figure 2.4 Sample of Compressed Text


There is, however, another line of attack. If the cryptanalyst knows the nature of the plaintext (e.g., noncompressed English text), then the analyst can exploit the regularities of the language. To see how such a cryptanalysis might proceed, we give a partial example here that is adapted from one in [SINK09]. The ciphertext to be solved is

UzqSovUoHxmoPvgPozPevSgzWSzoPfPeSxUDBmeTSxaIz

vUePHzHmDzSHzoWSfPaPPDTSvPqUzWymxUzUHSx

ePyePoPDzSzUfPomBzWPfUPzHmDJUDTmoHmq

As a first step, the relative frequency of the letters can be determined and compared to a standard frequency distribution for English, such as is shown in Figure 2.5 (based on [LEWA00]). If the message were long enough, this technique alone might be sufficient, but because this is a relatively short message, we cannot expect an exact match. In any case, the relative frequencies of the letters in the ciphertext (in percentages) are as follows:

P 13.33 H 5.83 F 3.33 B 1.67 C 0.00Z 11.67 D 5.00 W 3.33 G 1.67 K 0.00S 8.33 E 5.00 Q 2.50 Y 1.67 L 0.00U 8.33 V 4.17 T 2.50 I 0.83 N 0.00O 7.50 X 4.17 A 1.67 J 0.83 R 0.00M 6.67

Comparing this breakdown with Figure 2.5, it seems likely that cipher letters P and Z are the equivalents of plain letters e and t, but it is not certain which is which. The letters S, U, O, M, and H are all of relatively high frequency and probably cor-respond to plain letters from the set {a, h, i, n, o, r, s}. The letters with the lowest frequencies (namely, A, B, G, Y, I, J) are likely included in the set {b, j, k, q, v, x, z}.

There are a number of ways to proceed at this point. We could make some ten-tative assignments and start to fill in the plaintext to see if it looks like a reasonable “skeleton” of a message. A more systematic approach is to look for other regularities. For example, certain words may be known to be in the text. Or we could look for repeating sequences of cipher letters and try to deduce their plaintext equivalents.

A powerful tool is to look at the frequency of two-letter combinations, known as digrams. A table similar to Figure 2.5 could be drawn up showing the relative fre-quency of digrams. The most common such digram is th. In our ciphertext, the most common digram is ZW, which appears three times. So we make the correspondence of Z with t and W with h. Then, by our earlier hypothesis, we can equate P with e. Now notice that the sequence ZWP appears in the ciphertext, and we can translate that sequence as “the.” This is the most frequent trigram (three-letter combination) in English, which seems to indicate that we are on the right track.

Next, notice the sequence ZWSZ in the first line. We do not know that these four letters form a complete word, but if they do, it is of the form th_t. If so, S equates with a.


So far, then, we have

UzqSovUoHxmoPvgPozPevSgzWSzoPfPeSxUDBmeTSxaIz

t a e e te a that e e a a

vUePHzHmDzSHzoWSfPaPPDTSvPqUzWymxUzUHSx

e t ta t ha e ee a e th t a

ePyePoPDzSzUfPomBzWPfUPzHmDJUDTmoHmq

e e e tat e the t

Only four letters have been identified, but already we have quite a bit of the message. Continued analysis of frequencies plus trial and error should easily yield a solution from this point. The complete plaintext, with spaces added between words, follows:

it was disclosed yesterday that several informal but

direct contacts have been made with political

representatives of the viet cong in moscow

Monoalphabetic ciphers are easy to break because they reflect the frequency data of the original alphabet. A countermeasure is to provide multiple substitutes,

0

2

4

6

8

10

12

14

A

8.16

7

1.49

2

2.78

2

4.25

3

12.7

02

2.22

8

2.01

5

6.09

4 6.99

6

0.15

3 0.77

2

4.02

5

2.40

6

6.74

9 7.50

7

1.92

9

0.09

5

5.98

7

6.32

7

9.05

6

2.75

8

0.97

8

2.36

0

0.15

0

1.97

4

0.07

4

B C D E F G H I J K L M N

Rel

ativ

e fr

eque

ncy

(%)

O P Q R S T U V W X Y Z

Figure 2.5 Relative Frequency of Letters in English Text


known as homophones, for a single letter. For example, the letter e could be as-signed a number of different cipher symbols, such as 16, 74, 35, and 21, with each homophone assigned to a letter in rotation or randomly. If the number of symbols assigned to each letter is proportional to the relative frequency of that letter, then single-letter frequency information is completely obliterated. The great mathemati-cian Carl Friedrich Gauss believed that he had devised an unbreakable cipher using homophones. However, even with homophones, each element of plaintext affects only one element of ciphertext, and multiple-letter patterns (e.g., digram frequen-cies) still survive in the ciphertext, making cryptanalysis relatively straightforward.

Two principal methods are used in substitution ciphers to lessen the extent to which the structure of the plaintext survives in the ciphertext: One approach is to encrypt multiple letters of plaintext, and the other is to use multiple cipher alpha-bets. We briefly examine each.

Playfair Cipher

The best-known multiple-letter encryption cipher is the Playfair, which treats digrams in the plaintext as single units and translates these units into ciphertext digrams.3

The Playfair algorithm is based on the use of a 5 * 5 matrix of letters con-structed using a keyword. Here is an example, solved by Lord Peter Wimsey in Dorothy Sayers’s Have His Carcase:4

M O N A RC H Y B DE F G I/J KL P Q S TU V W X Z

In this case, the keyword is monarchy. The matrix is constructed by filling in the letters of the keyword (minus duplicates) from left to right and from top to bottom, and then filling in the remainder of the matrix with the remaining letters in alphabetic order. The letters I and J count as one letter. Plaintext is encrypted two letters at a time, according to the following rules:

1. Repeating plaintext letters that are in the same pair are separated with a filler letter, such as x, so that balloon would be treated as ba lx lo on.

2. Two plaintext letters that fall in the same row of the matrix are each replaced by the letter to the right, with the first element of the row circularly following the last. For example, ar is encrypted as RM.

3. Two plaintext letters that fall in the same column are each replaced by the letter beneath, with the top element of the column circularly following the last. For example, mu is encrypted as CM.

3This cipher was actually invented by British scientist Sir Charles Wheatstone in 1854, but it bears the name of his friend Baron Playfair of St. Andrews, who championed the cipher at the British foreign office.4The book provides an absorbing account of a probable-word attack.


4. Otherwise, each plaintext letter in a pair is replaced by the letter that lies in its own row and the column occupied by the other plaintext letter. Thus, hs becomes BP and ea becomes IM (or JM, as the encipherer wishes).

The Playfair cipher is a great advance over simple monoalphabetic ciphers. For one thing, whereas there are only 26 letters, there are 26 * 26 = 676 digrams, so that identification of individual digrams is more difficult. Furthermore, the relative frequencies of individual letters exhibit a much greater range than that of digrams, making frequency analysis much more difficult. For these reasons, the Playfair cipher was for a long time considered unbreakable. It was used as the standard field system by the British Army in World War I and still enjoyed considerable use by the U.S. Army and other Allied forces during World War II.

Despite this level of confidence in its security, the Playfair cipher is relatively easy to break, because it still leaves much of the structure of the plaintext language intact. A few hundred letters of ciphertext are generally sufficient.

One way of revealing the effectiveness of the Playfair and other ciphers is shown in Figure 2.6. The line labeled plaintext plots a typical frequency distribution of the 26 alphabetic characters (no distinction between upper and lower case) in ordinary text. This is also the frequency distribution of any monoalphabetic substitution cipher, because the frequency values for individual letters are the same, just with different letters substituted for the original letters. The plot is developed in the following way: The number of occurrences of each letter in the text is counted and divided by the number of occurrences of the most frequently used letter. Using the results of Figure 2.5, we see that e is the most frequently used letter. As a result, e has a relative frequency of 1, t of

0

1 2 3 4 5 6 1 7 8 9 10 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Plaintext

Playfair

Vignere

Random polyalphabetic

Frequency ranked letters (decreasing frequency)

Nor

mal

ized

rel

ativ

e fr

eque

ncy

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Figure 2.6 Relative Frequency of Occurrence of Letters


9.056/12.702 ≈ 0.72, and so on. The points on the horizontal axis correspond to the letters in order of decreasing frequency.

Figure 2.6 also shows the frequency distribution that results when the text is encrypted using the Playfair cipher. To normalize the plot, the number of occurrences of each letter in the ciphertext was again divided by the number of occurrences of e in the plaintext. The resulting plot therefore shows the extent to which the frequency distribution of letters, which makes it trivial to solve substitution ciphers, is masked by encryption. If the frequency distribution information were totally concealed in the encryption process, the ciphertext plot of frequencies would be flat, and cryptanalysis using ciphertext only would be effectively impossible. As the figure shows, the Playfair cipher has a flatter dis-tribution than does plaintext, but nevertheless, it reveals plenty of structure for a cryptanalyst to work with. The plot also shows the Vigenère cipher, discussed subsequently. The Hill and Vigenère curves on the plot are based on results reported in [SIMM93].

Hill Cipher5

Another interesting multiletter cipher is the Hill cipher, developed by the math-ematician Lester Hill in 1929.

ConCepts from Linear aLgebra Before describing the Hill cipher, let us briefly review some terminology from linear algebra. In this discussion, we are concerned with matrix arithmetic modulo 26. For the reader who needs a refresher on matrix multiplication and inversion, see Appendix E.

We define the inverse M- 1 of a square matrix M by the equation M(M- 1) = M- 1M = I, where I is the identity matrix. I is a square matrix that is all zeros except for ones along the main diagonal from upper left to lower right. The inverse of a matrix does not always exist, but when it does, it satisfies the preceding equation. For example,

A = a 5 817 3

b A- 1 mod 26 = a9 21 15

b

AA- 1 = a (5 * 9) + (8 * 1) (5 * 2) + (8 * 15)(17 * 9) + (3 * 1) (17 * 2) + (3 * 15)

b

= a 53 130156 79

b mod 26 = a1 00 1

b

To explain how the inverse of a matrix is computed, we begin with the concept of determinant. For any square matrix (m * m), the determinant equals the sum of all the products that can be formed by taking exactly one element from each row

5This cipher is somewhat more difficult to understand than the others in this chapter, but it illustrates an important point about cryptanalysis that will be useful later on. This subsection can be skipped on a first reading.


and exactly one element from each column, with certain of the product terms pre-ceded by a minus sign. For a 2 * 2 matrix,

ak11 k12

k21 k22b

the determinant is k11k22 - k12k21. For a 3 * 3 matrix, the value of the determi-nant is k11k22k33 + k21k32k13 + k31k12k23 - k31k22k13 - k21k12k33 - k11k32k23. If a square matrix A has a nonzero determinant, then the inverse of the matrix is com-puted as [A-1]ij = (det A)-1(-1)i+ j(Dji), where (Dji) is the subdeterminant formed by deleting the jth row and the ith column of A, det(A) is the determinant of A, and (det A)- 1 is the multiplicative inverse of (det A) mod 26.

Continuing our example,

det a 5 817 3

b = (5 * 3) - (8 * 17) = -121 mod 26 = 9

We can show that 9- 1 mod 26 = 3, because 9 * 3 = 27 mod 26 = 1 (see Chapter 4 or Appendix E). Therefore, we compute the inverse of A as

A = a 5 817 3

b

A- 1 mod 26 = 3 a 3 -8-17 5

b = 3 a3 189 5

b = a 9 5427 15

b = a9 21 15

b

the hiLL aLgorithm This encryption algorithm takes m successive plaintext let-ters and substitutes for them m ciphertext letters. The substitution is determined by m linear equations in which each character is assigned a numerical value (a = 0, b = 1, c, z = 25). For m = 3, the system can be described as

c1 = (k11p1 + k21p2 + k31p3) mod 26

c2 = (k12p1 + k22p2 + k32p3) mod 26

c3 = (k13p1 + k23p2 + k33p3) mod 26

This can be expressed in terms of row vectors and matrices:6

(c1 c2 c3) = (p1 p2 p3)£k11 k12 k13

k21 k22 k23

k31 k32 k33

≥ mod 26

or

C = PK mod 26

6Some cryptography books express the plaintext and ciphertext as column vectors, so that the column vector is placed after the matrix rather than the row vector placed before the matrix. Sage uses row vec-tors, so we adopt that convention.


where C and P are row vectors of length 3 representing the plaintext and ciphertext, and K is a 3 * 3 matrix representing the encryption key. Operations are performed mod 26.

For example, consider the plaintext “paymoremoney” and use the encryp-tion key

K = £17 17 521 18 212 2 19

≥The first three letters of the plaintext are represented by the vector (15 0 24). Then(15 0 24)K = (303 303 531) mod 26 = (17 17 11) = RRL. Continuing in this fashion, the ciphertext for the entire plaintext is RRLMWBKASPDH.

Decryption requires using the inverse of the matrix K. We can compute det K = 23, and therefore, (det K)-1 mod 26 = 17. We can then compute the inverse as7

K-1 = £ 4 9 1515 17 6 24 0 17

≥This is demonstrated as£17 17 5

21 18 21 2 2 19

≥£ 4 9 15 15 17 6 24 0 17

≥ = £443 442 442 858 495 780 494 52 365

≥ mod 26 = £1 0 0 0 1 0 0 0 1

≥It is easily seen that if the matrix K- 1 is applied to the ciphertext, then the

plaintext is recovered.In general terms, the Hill system can be expressed as

C = E(K, P) = PK mod 26

P = D(K, C) = CK- 1 mod 26 = PKK- 1 = P

As with Playfair, the strength of the Hill cipher is that it completely hides single-letter frequencies. Indeed, with Hill, the use of a larger matrix hides more frequency information. Thus, a 3 * 3 Hill cipher hides not only single-letter but also two-letter frequency information.

Although the Hill cipher is strong against a ciphertext-only attack, it is easily broken with a known plaintext attack. For an m * m Hill cipher, sup-pose we have m plaintext–ciphertext pairs, each of length m. We label the pairs Pj = (p1j p1j P pmj) and Cj = (c1j c1j P cmj) such that Cj = PjK for 1 … j … m and for some unknown key matrix K. Now define two m * m matrices X = (pij) and Y = (cij). Then we can form the matrix equation Y = XK. If X has an inverse, then we can determine K = X- 1Y. If X is not invertible, then a new version of X can be formed with additional plaintext–ciphertext pairs until an invertible X is obtained.

7The calculations for this example are provided in detail in Appendix E.


Consider this example. Suppose that the plaintext “hillcipher” is encrypted using a 2 * 2 Hill cipher to yield the ciphertext HCRZSSXNSP. Thus, we know that (7 8)K mod 26 = (7 2); (11 11)K mod 26 = (17 25); and so on. Using the first two plaintext–ciphertext pairs, we have

a 7 217 25

b = a 7 811 11

bK mod 26

The inverse of X can be computed:

a 7 811 11

b- 1

= a25 221 23

b

so

K = a25 221 23

ba 7 217 25

b = a549 600398 577

b mod 26 = a3 28 5

b

This result is verified by testing the remaining plaintext–ciphertext pairs.

Polyalphabetic Ciphers

Another way to improve on the simple monoalphabetic technique is to use differ-ent monoalphabetic substitutions as one proceeds through the plaintext message. The general name for this approach is polyalphabetic substitution cipher. All these techniques have the following features in common:

1. A set of related monoalphabetic substitution rules is used.

2. A key determines which particular rule is chosen for a given transformation.

Vigenère Cipher The best known, and one of the simplest, polyalphabetic ciphers is the Vigenère cipher. In this scheme, the set of related monoalphabetic substitu-tion rules consists of the 26 Caesar ciphers with shifts of 0 through 25. Each cipher is denoted by a key letter, which is the ciphertext letter that substitutes for the plaintext letter a. Thus, a Caesar cipher with a shift of 3 is denoted by the key value 3.8

We can express the Vigenère cipher in the following manner. Assume a sequence of plaintext letters P = p0, p1, p2, c, pn - 1 and a key consisting of the sequence of letters K = k0, k1, k2, c, km - 1, where typically m 6 n. The sequence of ciphertext letters C = C0, C1, C2, c, Cn - 1 is calculated as follows:

C = C0, C1, C2, c, Cn - 1 = E(K, P) = E[(k0, k1, k2, c, km - 1), (p0, p1, p2, c, pn - 1)] = (p0 + k0) mod 26, (p1 + k1) mod 26, c, (pm - 1 + km - 1) mod 26, (pm + k0) mod 26, (pm + 1 + k1) mod 26, c, (p2m - 1 + km - 1) mod 26, c

Thus, the first letter of the key is added to the first letter of the plaintext, mod 26, the second letters are added, and so on through the first m letters of the plaintext. For the next m letters of the plaintext, the key letters are repeated. This process

8To aid in understanding this scheme and also to aid in it use, a matrix known as the Vigenère tableau is often used. This tableau is discussed in a document in the Premium Content Web site for this book.


continues until all of the plaintext sequence is encrypted. A general equation of the encryption process is

Ci = (pi + ki mod m) mod 26 (2.3)

Compare this with Equation (2.1) for the Caesar cipher. In essence, each plaintext character is encrypted with a different Caesar cipher, depending on the corresponding key character. Similarly, decryption is a generalization of Equation (2.2):

pi = (Ci - ki mod m) mod 26 (2.4)

To encrypt a message, a key is needed that is as long as the message. Usually, the key is a repeating keyword. For example, if the keyword is deceptive, the message “we are discovered save yourself” is encrypted as

key: deceptivedeceptivedeceptive

plaintext: wearediscoveredsaveyourself

ciphertext: zIcvTWqngRzgvTWavzHcqyglmgJ

Expressed numerically, we have the following result.

key 3 4 2 4 15 19 8 21 4 3 4 2 4 15plaintext 22 4 0 17 4 3 8 18 2 14 21 4 17 4ciphertext 25 8 2 21 19 22 16 13 6 17 25 6 21 19

key 19 8 21 4 3 4 2 4 15 19 8 21 4plaintext 3 18 0 21 4 24 14 20 17 18 4 11 5ciphertext 22 0 21 25 7 2 16 24 6 11 12 6 9

The strength of this cipher is that there are multiple ciphertext letters for each plaintext letter, one for each unique letter of the keyword. Thus, the letter frequency information is obscured. However, not all knowledge of the plaintext structure is lost. For example, Figure 2.6 shows the frequency distribution for a Vigenère cipher with a keyword of length 9. An improvement is achieved over the Playfair cipher, but considerable frequency information remains.

It is instructive to sketch a method of breaking this cipher, because the method reveals some of the mathematical principles that apply in cryptanalysis.

First, suppose that the opponent believes that the ciphertext was encrypted using either monoalphabetic substitution or a Vigenère cipher. A simple test can be made to make a determination. If a monoalphabetic substitution is used, then the statistical properties of the ciphertext should be the same as that of the lan-guage of the plaintext. Thus, referring to Figure 2.5, there should be one cipher letter with a relative frequency of occurrence of about 12.7%, one with about 9.06%, and so on. If only a single message is available for analysis, we would not expect an exact match of this small sample with the statistical profile of the plaintext language. Nevertheless, if the correspondence is close, we can assume a monoalphabetic substitution.


If, on the other hand, a Vigenère cipher is suspected, then progress depends on determining the length of the keyword, as will be seen in a moment. For now, let us concentrate on how the keyword length can be determined. The important in-sight that leads to a solution is the following: If two identical sequences of plaintext letters occur at a distance that is an integer multiple of the keyword length, they will generate identical ciphertext sequences. In the foregoing example, two instances of the sequence “red” are separated by nine character positions. Consequently, in both cases, r is encrypted using key letter e, e is encrypted using key letter p, and d is encrypted using key letter t. Thus, in both cases, the ciphertext sequence is VTW. We indicate this above by underlining the relevant ciphertext letters and shading the relevant ciphertext numbers.

An analyst looking at only the ciphertext would detect the repeated sequences VTW at a displacement of 9 and make the assumption that the keyword is either three or nine letters in length. The appearance of VTW twice could be by chance and may not reflect identical plaintext letters encrypted with identical key letters. However, if the message is long enough, there will be a number of such repeated ciphertext sequences. By looking for common factors in the displacements of the various sequences, the analyst should be able to make a good guess of the keyword length.

Solution of the cipher now depends on an important insight. If the keyword length is m, then the cipher, in effect, consists of m monoalphabetic substitution ciphers. For example, with the keyword DECEPTIVE, the letters in positions 1, 10, 19, and so on are all encrypted with the same monoalphabetic cipher. Thus, we can use the known frequency characteristics of the plaintext language to attack each of the monoalphabetic ciphers separately.

The periodic nature of the keyword can be eliminated by using a nonrepeating keyword that is as long as the message itself. Vigenère proposed what is referred to as an autokey system, in which a keyword is concatenated with the plaintext itself to provide a running key. For our example,

key: deceptivewearediscoveredsav

plaintext: wearediscoveredsaveyourself

ciphertext: zIcvTWqngKzeIIgaSxSTSlvvWla

Even this scheme is vulnerable to cryptanalysis. Because the key and the plaintext share the same frequency distribution of letters, a statistical technique can be applied. For example, e enciphered by e, by Figure 2.5, can be expected to occur with a frequency of (0.127)2 ≈ 0.016, whereas t enciphered by t would occur only about half as often. These regularities can be exploited to achieve successful cryptanalysis.9

Vernam Cipher The ultimate defense against such a cryptanalysis is to choose a keyword that is as long as the plaintext and has no statistical relationship to it. Such a system was introduced by an AT&T engineer named Gilbert Vernam in 1918.

9Although the techniques for breaking a Vigenère cipher are by no means complex, a 1917 issue of Scientific American characterized this system as “impossible of translation.” This is a point worth remem-bering when similar claims are made for modern algorithms.


His system works on binary data (bits) rather than letters. The system can be expressed succinctly as follows (Figure 2.7):

ci = pi ⊕ ki

where

pi = ith binary digit of plaintext

ki = ith binary digit of key

ci = ith binary digit of ciphertext

⊕ = exclusive-or (XOR) operation

Compare this with Equation (2.3) for the Vigenère cipher.Thus, the ciphertext is generated by performing the bitwise XOR of the plain-

text and the key. Because of the properties of the XOR, decryption simply involves the same bitwise operation:

pi = ci ⊕ ki

which compares with Equation (2.4).The essence of this technique is the means of construction of the key. Vernam

proposed the use of a running loop of tape that eventually repeated the key, so that in fact the system worked with a very long but repeating keyword. Although such a scheme, with a long key, presents formidable cryptanalytic difficulties, it can be broken with sufficient ciphertext, the use of known or probable plaintext sequences, or both.

One-Time Pad

An Army Signal Corp officer, Joseph Mauborgne, proposed an improvement to the Vernam cipher that yields the ultimate in security. Mauborgne suggested using a random key that is as long as the message, so that the key need not be repeated. In addition, the key is to be used to encrypt and decrypt a single message, and then is discarded. Each new message requires a new key of the same length as the new mes-sage. Such a scheme, known as a one-time pad, is unbreakable. It produces random output that bears no statistical relationship to the plaintext. Because the ciphertext contains no information whatsoever about the plaintext, there is simply no way to break the code.

Key streamgenerator

Cryptographicbit stream ( ki )

Cryptographicbit stream ( ki )

Plaintext( pi )

Plaintext( pi )

Ciphertext( ci )

Key streamgenerator

Figure 2.7 Vernam Cipher


An example should illustrate our point. Suppose that we are using a Vigenère scheme with 27 characters in which the twenty-seventh character is the space character, but with a one-time key that is as long as the message. Consider the ciphertext

anKyoDKyURePfJByoJDSPlReyIUnofDoIUeRfPlUyTS

We now show two different decryptions using two different keys:

ciphertext: anKyoDKyURePfJByoJDSPlReyIUnofDoIUeRfPlUyTS

key: pxlmvmsydofuyrvzwc tnlebnecvgdupahfzzlmnyih

plaintext: mr mustard with the candlestick in the hall

ciphertext: anKyoDKyURePfJByoJDSPlReyIUnofDoIUeRfPlUyTS

key: pftgpmiydgaxgoufhklllmhsqdqogtewbqfgyovuhwt

plaintext: miss scarlet with the knife in the library

Suppose that a cryptanalyst had managed to find these two keys. Two plau-sible plaintexts are produced. How is the cryptanalyst to decide which is the correct decryption (i.e., which is the correct key)? If the actual key were produced in a truly random fashion, then the cryptanalyst cannot say that one of these two keys is more likely than the other. Thus, there is no way to decide which key is correct and there-fore which plaintext is correct.

In fact, given any plaintext of equal length to the ciphertext, there is a key that produces that plaintext. Therefore, if you did an exhaustive search of all possible keys, you would end up with many legible plaintexts, with no way of knowing which was the intended plaintext. Therefore, the code is unbreakable.

The security of the one-time pad is entirely due to the randomness of the key. If the stream of characters that constitute the key is truly random, then the stream of characters that constitute the ciphertext will be truly random. Thus, there are no patterns or regularities that a cryptanalyst can use to attack the ciphertext.

In theory, we need look no further for a cipher. The one-time pad offers com-plete security but, in practice, has two fundamental difficulties:

1. There is the practical problem of making large quantities of random keys. Any heavily used system might require millions of random characters on a regular basis. Supplying truly random characters in this volume is a significant task.

2. Even more daunting is the problem of key distribution and protection. For every message to be sent, a key of equal length is needed by both sender and receiver. Thus, a mammoth key distribution problem exists.

Because of these difficulties, the one-time pad is of limited utility and is useful primarily for low-bandwidth channels requiring very high security.

The one-time pad is the only cryptosystem that exhibits what is referred to as perfect secrecy. This concept is explored in Appendix F.

2.3 / transposition teChniques 49

2.3 transpOsitiOn techniques

All the techniques examined so far involve the substitution of a ciphertext symbol for a plaintext symbol. A very different kind of mapping is achieved by performing some sort of permutation on the plaintext letters. This technique is referred to as a transposition cipher.

The simplest such cipher is the rail fence technique, in which the plaintext is written down as a sequence of diagonals and then read off as a sequence of rows. For example, to encipher the message “meet me after the toga party” with a rail fence of depth 2, we write the following:

m e m a t r h t g p r y

e t e f e t e o a a t

The encrypted message is

MEMATRHTGPRYETEFETEOAAT

This sort of thing would be trivial to cryptanalyze. A more complex scheme is to write the message in a rectangle, row by row, and read the message off, column by column, but permute the order of the columns. The order of the columns then becomes the key to the algorithm. For example,

Key: 4 3 1 2 5 6 7

Plaintext: a t t a c k p

o s t p o n e

d u n t i l t

w o a m x y z

ciphertext: TTnaaPTmTSUoaoDWcoIxKnlyPeTz

Thus, in this example, the key is 4312567. To encrypt, start with the column that is labeled 1, in this case column 3. Write down all the letters in that column. Proceed to column 4, which is labeled 2, then column 2, then column 1, then columns 5, 6, and 7.

A pure transposition cipher is easily recognized because it has the same letter frequencies as the original plaintext. For the type of columnar transposition just shown, cryptanalysis is fairly straightforward and involves laying out the cipher-text in a matrix and playing around with column positions. Digram and trigram frequency tables can be useful.

The transposition cipher can be made significantly more secure by perform-ing more than one stage of transposition. The result is a more complex permutation that is not easily reconstructed. Thus, if the foregoing message is reencrypted using the same algorithm,


Key: 4 3 1 2 5 6 7

Input: t t n a a p t

m t s u o a o

d w c o i x k

n l y p e t z

output: nScyaUoPTTWlTmDnaoIePaxTToKz

To visualize the result of this double transposition, designate the letters in the original plaintext message by the numbers designating their position. Thus, with 28 letters in the message, the original sequence of letters is

01 02 03 04 05 06 07 08 09 10 11 12 13 14

15 16 17 18 19 20 21 22 23 24 25 26 27 28

After the first transposition, we have

03 10 17 24 04 11 18 25 02 09 16 23 01 08

15 22 05 12 19 26 06 13 20 27 07 14 21 28

which has a somewhat regular structure. But after the second transposition, we have

17 09 05 27 24 16 12 07 10 02 22 20 03 25

15 13 04 23 19 14 11 01 26 21 18 08 06 28

This is a much less structured permutation and is much more difficult to cryptanalyze.

2.4 rOtOr machines

The example just given suggests that multiple stages of encryption can produce an algorithm that is significantly more difficult to cryptanalyze. This is as true of substi-tution ciphers as it is of transposition ciphers. Before the introduction of DES, the most important application of the principle of multiple stages of encryption was a class of systems known as rotor machines.10

The basic principle of the rotor machine is illustrated in Figure 2.8. The ma-chine consists of a set of independently rotating cylinders through which electrical pulses can flow. Each cylinder has 26 input pins and 26 output pins, with internal wiring that connects each input pin to a unique output pin. For simplicity, only three of the internal connections in each cylinder are shown.

If we associate each input and output pin with a letter of the alphabet, then a single cylinder defines a monoalphabetic substitution. For example, in Figure 2.8, if an operator depresses the key for the letter A, an electric signal is applied to

10Machines based on the rotor principle were used by both Germany (Enigma) and Japan (Purple) in World War II. The breaking of both codes by the Allies was a significant factor in the war’s outcome.

51

2425261234567891011121314151617181920212223

213

151

19101426208

167

224

115

179

1223182

256

2413

ABCDEFGHIJ

KL

MNOPQRSTUVWXYZ

2612345678910111213141516171819202122232425

20164

153

1412235

162

221911182524137

108

219

2617

1234567891011121314151617181920212223242526

81826172022103

13114

235

249

122516196

1521271

14

ABCDEFGHIJ

KL

MNOPQRSTUVWXYZ

Direction of motion Direction of motion

Fast rotor Medium rotor Slow rotor Fast rotor Medium rotor Slow rotor

(a) Initial setting (b) Setting after one keystroke

ABCDEFGHIJ

KL

MNOPQRSTUVWXYZ

2324252612345678910111213141516171819202122

13213

151

19101426208

167

224

115

179

1223182

256

24

2612345678910111213141516171819202122232425

20164

153

1412235

162

221911182524137

108

219

2617

1234567891011121314151617181920212223242526

81826172022103

13114

235

249

122516196

1521271

14

ABCDEFGHIJ

KL

MNOPQRSTUVWXYZ

Figure 2.8 Three-Rotor Machine with Wiring Represented by Numbered Contacts


the first pin of the first cylinder and flows through the internal connection to the twenty-fifth output pin.

Consider a machine with a single cylinder. After each input key is depressed, the cylinder rotates one position, so that the internal connections are shifted accordingly. Thus, a different monoalphabetic substitution cipher is defined. After 26 letters of plaintext, the cylinder would be back to the initial position. Thus, we have a polyalphabetic substitution algorithm with a period of 26.

A single-cylinder system is trivial and does not present a formidable cryptana-lytic task. The power of the rotor machine is in the use of multiple cylinders, in which the output pins of one cylinder are connected to the input pins of the next. Figure 2.8 shows a three-cylinder system. The left half of the figure shows a position in which the input from the operator to the first pin (plaintext letter a) is routed through the three cylinders to appear at the output of the second pin (ciphertext letter B).

With multiple cylinders, the one closest to the operator input rotates one pin position with each keystroke. The right half of Figure 2.8 shows the system’s configuration after a single keystroke. For every complete rotation of the inner cylinder, the middle cylinder rotates one pin position. Finally, for every complete rotation of the middle cylinder, the outer cylinder rotates one pin position. This is the same type of operation seen with an odometer. The result is that there are 26 * 26 * 26 = 17,576 different substitution alphabets used before the system repeats. The addition of fourth and fifth rotors results in periods of 456,976 and 11,881,376 letters, respectively. Thus, a given setting of a 5-rotor machine is equiva-lent to a Vigenère cipher with a key length of 11,881,376.

Such a scheme presents a formidable cryptanalytic challenge. If, for example, the cryptanalyst attempts to use a letter frequency analysis approach, the analyst is faced with the equivalent of over 11 million monoalphabetic ciphers. We might need on the order of 50 letters in each monalphabetic cipher for a solution, which means that the analyst would need to be in possession of a ciphertext with a length of over half a billion letters.

The significance of the rotor machine today is that it points the way to the most widely used cipher ever: the Data Encryption Standard (DES), which is intro-duced in Chapter 3.

2.5 steganOgraphy

We conclude with a discussion of a technique that (strictly speaking), is not encryp-tion, namely, steganography.

A plaintext message may be hidden in one of two ways. The methods of steganography conceal the existence of the message, whereas the methods of cryptography render the message unintelligible to outsiders by various transfor-mations of the text.11

11Steganography was an obsolete word that was revived by David Kahn and given the meaning it has today [KAHN96].

2.5 / steganography 53

A simple form of steganography, but one that is time-consuming to con-struct, is one in which an arrangement of words or letters within an appar-ently innocuous text spells out the real message. For example, the sequence of first letters of each word of the overall message spells out the hidden message. Figure 2.9 shows an example in which a subset of the words of the overall mes-sage is used to convey the hidden message. See if you can decipher this; it’s not too hard.

Various other techniques have been used historically; some examples are the following [MYER91]:

• Character marking: Selected letters of printed or typewritten text are over-written in pencil. The marks are ordinarily not visible unless the paper is held at an angle to bright light.

• Invisible ink: A number of substances can be used for writing but leave no visible trace until heat or some chemical is applied to the paper.

• Pin punctures: Small pin punctures on selected letters are ordinarily not visible unless the paper is held up in front of a light.

• Typewriter correction ribbon: Used between lines typed with a black ribbon, the results of typing with the correction tape are visible only under a strong light.

Figure 2.9 A Puzzle for Inspector Morse(From The Silent World of Nicholas Quinn, by Colin Dexter)


Although these techniques may seem archaic, they have contemporary equiv-alents. [WAYN09] proposes hiding a message by using the least significant bits of frames on a CD. For example, the Kodak Photo CD format’s maximum resolution is 3096 * 6144 pixels, with each pixel containing 24 bits of RGB color information. The least significant bit of each 24-bit pixel can be changed without greatly affecting the quality of the image. The result is that you can hide a 130-kB message in a single digital snapshot. There are now a number of software packages available that take this type of approach to steganography.

Steganography has a number of drawbacks when compared to encryption. It requires a lot of overhead to hide a relatively few bits of information, although using a scheme like that proposed in the preceding paragraph may make it more effective. Also, once the system is discovered, it becomes virtually worthless. This problem, too, can be overcome if the insertion method depends on some sort of key (e.g., see Problem 2.20). Alternatively, a message can be first encrypted and then hidden using steganography.

The advantage of steganography is that it can be employed by parties who have something to lose should the fact of their secret communication (not necessar-ily the content) be discovered. Encryption flags traffic as important or secret or may identify the sender or receiver as someone with something to hide.


For anyone interested in the history of code making and code breaking, the book to read is [KAHN96]. Although it is concerned more with the impact of cryptology than its technical development, it is an excellent introduction and makes for exciting reading. Another excel-lent historical account is [SING99].

A short treatment covering the techniques of this chapter, and more, is [GARD72]. There are many books that cover classical cryptography in a more technical vein; one of the best is [SINK09]. [KORN96] is a delightful book to read and contains a lengthy section on classical techniques. Two cryptography books that contain a fair amount of technical mate-rial on classical techniques are [GARR01] and [NICH99]. For the truly interested reader, the two-volume [NICH96] covers numerous classical ciphers in detail and provides many ciphertexts to be cryptanalyzed, together with the solutions.

An excellent treatment of rotor machines, including a discussion of their cryptanalysis is found in [KUMA97].

[KATZ00] provides a thorough treatment of steganography. Another good source is [WAYN09].

GARD72 Gardner, M. Codes, Ciphers, and Secret Writing. New York: Dover, 1972.GARR01 Garrett, P. Making, Breaking Codes: An Introduction to Cryptology. Upper

Saddle River, NJ: Prentice Hall, 2001.KAHN96 Kahn, D. The Codebreakers: The Story of Secret Writing. New York:

Scribner, 1996.KATZ00 Katzenbeisser, S., ed. Information Hiding Techniques for Steganography and

Digital Watermarking. Boston: Artech House, 2000.

2.7 / Key terms, review questions, and problems 55

2.7 Key terms, review questiOns, and prObLems

Key Terms

block cipherbrute-force attackCaesar ciphercipherciphertextcomputationally secureconventional encryptioncryptanalysiscryptographic systemcryptography

cryptologydecipheringdecryptiondigramencipheringencryptionHill ciphermonoalphabetic cipherone-time padplaintext

Playfair cipherpolyalphabetic cipherrail fence ciphersingle-key encryptionsteganographystream ciphersymmetric encryptiontransposition cipherunconditionally secureVigenère cipher

Review Questions 2.1 What are the essential ingredients of a symmetric cipher? 2.2 What are the two basic functions used in encryption algorithms? 2.3 How many keys are required for two people to communicate via a cipher? 2.4 What is the difference between a block cipher and a stream cipher? 2.5 What are the two general approaches to attacking a cipher? 2.6 List and briefly define types of cryptanalytic attacks based on what is known to the

attacker. 2.7 What is the difference between an unconditionally secure cipher and a computation-

ally secure cipher? 2.8 Briefly define the Caesar cipher. 2.9 Briefly define the monoalphabetic cipher. 2.10 Briefly define the Playfair cipher.

KORN96 Korner, T. The Pleasures of Counting. Cambridge, England: Cambridge University Press, 1996.

KUMA97 Kumar, I. Cryptology. Laguna Hills, CA: Aegean Park Press, 1997.NICH96 Nichols, R. Classical Cryptography Course. Laguna Hills, CA: Aegean Park

Press, 1996.NICH99 Nichols, R., ed. ICSA Guide to Cryptography. New York: McGraw-Hill, 1999.SING99 Singh, S. The Code Book: The Science of Secrecy from Ancient Egypt to

Quantum Cryptography. New York: Anchor Books, 1999.SINK09 Sinkov, A., and Feil, T. Elementary Cryptanalysis: A Mathematical Approach.

Washington, D.C.: The Mathematical Association of America, 2009.WAYN09 Wayner, P. Disappearing Cryptography. Boston: AP Professional Books,

2009.


2.11 What is the difference between a monoalphabetic cipher and a polyalphabetic cipher? 2.12 What are two problems with the one-time pad? 2.13 What is a transposition cipher? 2.14 What is steganography?

Problems 2.1 A generalization of the Caesar cipher, known as the affine Caesar cipher, has the

following form: For each plaintext letter p, substitute the ciphertext letter C:

C = E([a, b], p) = (ap + b) mod 26

A basic requirement of any encryption algorithm is that it be one-to-one. That is, if p ≠ q, then E(k, p) ≠ E(k, q). Otherwise, decryption is impossible, because more than one plaintext character maps into the same ciphertext character. The affine Caesar cipher is not one-to-one for all values of a. For example, for a = 2 and b = 3, then E([a, b], 0) = E([a, b], 13) = 3.

a. Are there any limitations on the value of b? Explain why or why not. b. Determine which values of a are not allowed. c. Provide a general statement of which values of a are and are not allowed. Justify

your statement. 2.2 How many one-to-one affine Caesar ciphers are there? 2.3 A ciphertext has been generated with an affine cipher. The most frequent letter of

the ciphertext is “B,” and the second most frequent letter of the ciphertext is “U.” Break this code.

2.4 The following ciphertext was generated using a simple substitution algorithm.

53‡‡†305))6*;4826)4‡.)4‡);806*;48†8¶60))85;;]8*;:‡*8†83(88)5*†;46(;88*96*?;8)*‡(;485);5*†2:*‡(;4956*2(5*—4)8¶8*;4069285);)6†8)4‡‡;1(‡9;48081;8:8‡1;48†85;4)485†528806*81(‡9;48;(88;4(‡?34;48)4‡;161;:188;‡?;

Decrypt this message.Hints:

1. As you know, the most frequently occurring letter in English is e. Therefore, the first or second (or perhaps third?) most common character in the message is likely to stand for e. Also, e is often seen in pairs (e.g., meet, fleet, speed, seen, been, agree, etc.). Try to find a character in the ciphertext that decodes to e.

2. The most common word in English is “the.” Use this fact to guess the characters that stand for t and h.

3. Decipher the rest of the message by deducing additional words.Warning: The resulting message is in English but may not make much sense on a first reading.

2.5 One way to solve the key distribution problem is to use a line from a book that both the sender and the receiver possess. Typically, at least in spy novels, the first sen-tence of a book serves as the key. The particular scheme discussed in this problem is from one of the best suspense novels involving secret codes, Talking to Strange Men, by Ruth Rendell. Work this problem without consulting that book!Consider the following message:

SIDKHKDM AF HCRKIABIE SHIMC KD LFEAILA


This ciphertext was produced using the first sentence of The Other Side of Silence (a book about the spy Kim Philby):

The snow lay thick on the steps and the snowflakes driven by the wind looked black in the headlights of the cars.

A simple substitution cipher was used. a. What is the encryption algorithm? b. How secure is it? c. To make the key distribution problem simple, both parties can agree to use the

first or last sentence of a book as the key. To change the key, they simply need to agree on a new book. The use of the first sentence would be preferable to the use of the last. Why?

2.6 In one of his cases, Sherlock Holmes was confronted with the following message.

534 C2 13 127 36 31 4 17 21 41DOUGLAS 109 293 5 37 BIRLSTONE

26 BIRLSTONE 9 127 171

Although Watson was puzzled, Holmes was able immediately to deduce the type of cipher. Can you?

2.7 This problem uses a real-world example, from an old U.S. Special Forces manual (public domain). The document, filename SpecialForces.pdf, is available at the Premium Content site for this book.

a. Using the two keys (memory words) cryptographic and network security, encrypt the following message:

Be at the third pillar from the left outside the lyceum theatre tonight at seven. If you are distrustful bring two friends.

Make reasonable assumptions about how to treat redundant letters and excess letters in the memory words and how to treat spaces and punctuation. Indicate what your assumptions are. Note: The message is from the Sherlock Holmes novel, The Sign of Four.

b. Decrypt the ciphertext. Show your work. c. Comment on when it would be appropriate to use this technique and what its

advantages are. 2.8 A disadvantage of the general monoalphabetic cipher is that both sender and receiver

must commit the permuted cipher sequence to memory. A common technique for avoiding this is to use a keyword from which the cipher sequence can be generated. For example, using the keyword CIPHER, write out the keyword followed by unused letters in normal order and match this against the plaintext letters:

plain: a b c d e f g h i j k l m n o p q r s t u v w x y zcipher: c I P H e R a B D f g J K l m n o q S T U v W x y z

If it is felt that this process does not produce sufficient mixing, write the remaining letters on successive lines and then generate the sequence by reading down the columns:

c I P H e Ra B D f g JK l m n o qS T U v W xy z


This yields the sequence:

c a K S y I B l T z P D m U H f n v e g o W R J q x

Such a system is used in the example in Section 2.2 (the one that begins “it was dis-closed yesterday”). Determine the keyword.

2.9 When the PT-109 American patrol boat, under the command of Lieutenant John F. Kennedy, was sunk by a Japanese destroyer, a message was received at an Australian wireless station in Playfair code:

KxJey UReBe zWeHe WRyTU HeyfSKReHe goyfI WTTTU olKSy caJPoBoTeI zonTx ByBnT goney cUzWRgDSon SxBoU yWRHe BaaHy USeDq

The key used was royal new zealand navy. Decrypt the message. Translate TT into tt. 2.10 a. Construct a Playfair matrix with the key largest. b. Construct a Playfair matrix with the key occurrence. Make a reasonable assump-

tion about how to treat redundant letters in the key. 2.11 a. Using this Playfair matrix:

M F H I/J K

U N O P Q

Z V W X Y

E L A R G

D S T B C

Encrypt this message:

Must see you over Cadogan West. Coming at once.

Note: The message is from the Sherlock Holmes story, The Adventure of the Bruce-Partington Plans.

b. Repeat part (a) using the Playfair matrix from Problem 2.10a. c. How do you account for the results of this problem? Can you generalize your

conclusion? 2.12 a. How many possible keys does the Playfair cipher have? Ignore the fact that some

keys might produce identical encryption results. Express your answer as an ap-proximate power of 2.

b. Now take into account the fact that some Playfair keys produce the same encryp-tion results. How many effectively unique keys does the Playfair cipher have?

2.13 What substitution system results when we use a 25 * 1 Playfair matrix? 2.14 a. Encrypt the message “meet me at the usual place at ten rather than eight oclock”

using the Hill cipher with the key a9 45 7

b . Show your calculations and the result.

b. Show the calculations for the corresponding decryption of the ciphertext to re-cover the original plaintext.

2.15 We have shown that the Hill cipher succumbs to a known plaintext attack if sufficient plaintext–ciphertext pairs are provided. It is even easier to solve the Hill cipher if a chosen plaintext attack can be mounted. Describe such an attack.

2.16 It can be shown that the Hill cipher with the matrix aa bc d

b requires that (ad - bc)

is relatively prime to 26; that is, the only common positive integer factor of (ad - bc) and 26 is 1. Thus, if (ad - bc) = 13 or is even, the matrix is not allowed. Determine


the number of different (good) keys there are for a 2 * 2 Hill cipher without counting them one by one, using the following steps:

a. Find the number of matrices whose determinant is even because one or both rows are even. (A row is “even” if both entries in the row are even.)

b. Find the number of matrices whose determinant is even because one or both columns are even. (A column is “even” if both entries in the column are even.)

c. Find the number of matrices whose determinant is even because all of the entries are odd.

d. Taking into account overlaps, find the total number of matrices whose determinant is even.

e. Find the number of matrices whose determinant is a multiple of 13 because the first column is a multiple of 13.

f. Find the number of matrices whose determinant is a multiple of 13 where the first column is not a multiple of 13 but the second column is a multiple of the first modulo 13.

g. Find the total number of matrices whose determinant is a multiple of 13. h. Find the number of matrices whose determinant is a multiple of 26 because they

fit cases parts (a) and (e), (b) and (e), (c) and (e), (a) and (f), and so on. i. Find the total number of matrices whose determinant is neither a multiple of 2 nor

a multiple of 13. 2.17 Calculate the determinant mod 26 of

a. a20 25 4

b b. °1 7 224 9 21 2 5

¢

2.18 Determine the inverse mod 26 of

a. a2 31 22

b b. °6 24 113 16 1020 17 15

¢

2.19 Using the Vigenère cipher, encrypt the word “explanation” using the key leg. 2.20 This problem explores the use of a one-time pad version of the Vigenère cipher.

In this scheme, the key is a stream of random numbers between 0 and 26. For example, if the key is 3 19 5 . . . , then the first letter of plaintext is encrypted with a shift of 3 letters, the second with a shift of 19 letters, the third with a shift of 5 letters, and so on.

a. Encrypt the plaintext sendmoremoney with the key stream

9 0 1 7 23 15 21 14 11 11 2 8 9

b. Using the ciphertext produced in part (a), find a key so that the cipher text decrypts to the plaintext cashnotneeded.

2.21 What is the message embedded in Figure 2.9?

Programming Problems 2.22 Write a program that can encrypt and decrypt using the general Caesar cipher, also

known as an additive cipher. 2.23 Write a program that can encrypt and decrypt using the affine cipher described in

Problem 2.1.


2.24 Write a program that can perform a letter frequency attack on an additive cipher without human intervention. Your software should produce possible plaintexts in rough order of likelihood. It would be good if your user interface allowed the user to specify “give me the top 10 possible plaintexts.”

2.25 Write a program that can perform a letter frequency attack on any monoalphabetic substitution cipher without human intervention. Your software should produce pos-sible plaintexts in rough order of likelihood. It would be good if your user interface allowed the user to specify “give me the top 10 possible plaintexts.”

2.26 Create software that can encrypt and decrypt using a 2 * 2 Hill cipher. 2.27 Create software that can perform a fast known plaintext attack on a Hill cipher, given

the dimension m. How fast are your algorithms, as a function of m?

61

3.1 Traditional Block Cipher Structure

Stream Ciphers and Block CiphersMotivation for the Feistel Cipher StructureThe Feistel Cipher

3.2 The Data Encryption Standard

DES EncryptionDES Decryption

3.3 A DES Example

ResultsThe Avalanche Effect

3.4 The Strength of DES

The Use of 56-Bit KeysThe Nature of the DES AlgorithmTiming Attacks

3.5 Block Cipher Design Principles

Number of RoundsDesign of Function FKey Schedule Algorithm



Chapter

Block ciphers and the data encryption standard

62 Chapter 3 / BloCk Ciphers and the data enCryption standard

“But what is the use of the cipher message without the cipher?”

—The Valley of Fear, Sir Arthur Conan Doyle

The objective of this chapter is to illustrate the principles of modern symmetric ciphers. For this purpose, we focus on the most widely used symmetric cipher: the Data Encryption Standard (DES). Although numerous symmetric ciphers have been developed since the introduction of DES, and although it is destined to be replaced by the Advanced Encryption Standard (AES), DES remains the most important such algorithm. Furthermore, a detailed study of DES provides an understanding of the principles used in other symmetric ciphers.

This chapter begins with a discussion of the general principles of symmetric block ciphers, which are the type of symmetric ciphers studied in this book (with the exception of the stream cipher RC4 in Chapter 7). Next, we cover full DES. Following this look at a specific algorithm, we return to a more general discussion of block cipher design.

Compared to public-key ciphers, such as RSA, the structure of DES and most symmetric ciphers is very complex and cannot be explained as easily as RSA and simi-lar algorithms. Accordingly, the reader may wish to begin with a simplified version of DES, which is described in Appendix G. This version allows the reader to perform encryption and decryption by hand and gain a good understanding of the working of the algorithm details. Classroom experience indicates that a study of this simplified version enhances understanding of DES.1

Learning Objectives

After studying this chapter, you should be able to

u Understand the distinction between stream ciphers and block ciphers.

u Present an overview of the Feistel cipher and explain how decryption is the inverse of encryption.

u Present an overview of Data Encryption Standard (DES).

u Explain the concept of the avalanche effect.

u Discuss the cryptographic strength of DES.

u Summarize the principal block cipher design principles.

1However, you may safely skip Appendix G, at least on a first reading. If you get lost or bogged down in the details of DES, then you can go back and start with simplified DES.

3.1 / traditional BloCk Cipher struCture 63

3.1 traditiOnaL bLOck cipher structure

Many symmetric block encryption algorithms in current use are based on a struc-ture referred to as a Feistel block cipher [FEIS73]. For that reason, it is important to examine the design principles of the Feistel cipher. We begin with a comparison of stream ciphers and block ciphers. Then we discuss the motivation for the Feistel block cipher structure. Finally, we discuss some of its implications.

Stream Ciphers and Block Ciphers

A stream cipher is one that encrypts a digital data stream one bit or one byte at a time. Examples of classical stream ciphers are the autokeyed Vigenère cipher and the Vernam cipher. In the ideal case, a one-time pad version of the Vernam cipher would be used (Figure 2.7), in which the keystream (ki) is as long as the plaintext bit stream (pi). If the cryptographic keystream is random, then this cipher is unbreakable by any means other than acquiring the keystream. However, the keystream must be provided to both users in advance via some independent and secure channel. This introduces insurmountable logistical problems if the intended data traffic is very large.

Accordingly, for practical reasons, the bit-stream generator must be implemented as an algorithmic procedure, so that the cryptographic bit stream can be produced by both users. In this approach (Figure 3.1a), the bit-stream generator is a key-controlled algorithm and must produce a bit stream that is cryptographically strong. That is, it must be computationally impractical to predict future portions of the bit stream based on previous portions of the bit stream. The two users need only share the generating key, and each can produce the keystream.

A block cipher is one in which a block of plaintext is treated as a whole and used to produce a ciphertext block of equal length. Typically, a block size of 64 or 128 bits is used. As with a stream cipher, the two users share a symmetric encryption key (Figure 3.1b). Using some of the modes of operation explained in Chapter 6, a block cipher can be used to achieve the same effect as a stream cipher.

Far more effort has gone into analyzing block ciphers. In general, they seem applicable to a broader range of applications than stream ciphers. The vast ma-jority of network-based symmetric cryptographic applications make use of block ciphers. Accordingly, the concern in this chapter, and in our discussions throughout the book of symmetric encryption, will primarily focus on block ciphers.

Motivation for the Feistel Cipher Structure

A block cipher operates on a plaintext block of n bits to produce a ciphertext block of n bits. There are 2n possible different plaintext blocks and, for the encryption to be reversible (i.e., for decryption to be possible), each must pro-duce a unique ciphertext block. Such a transformation is called reversible, or


nonsingular. The following examples illustrate nonsingular and singular transfor-mations for n = 2.

Bit-streamgenerationalgorithm

ENCRYPTION

(a) Stream cipher using algorithmic bit-stream generator

(b) Block cipher

Key( K )

Encryptionalgorithm

Plaintext

b bits

b bits

Key( K )

ki

Plaintext(pi)

Plaintext(pi)

Bit-streamgenerationalgorithm

DECRYPTION

Key( K )

ki

Ciphertext(ci)

Ciphertext

Decryptionalgorithm

Ciphertext

b bits

b bits

Key( K )

Plaintext

Figure 3.1 Stream Cipher and Block Cipher

2The reasoning is as follows: For the first plaintext, we can choose any of 2n ciphertext blocks. For the second plaintext, we choose from among 2n - 1 remaining ciphertext blocks, and so on.

In the latter case, a ciphertext of 01 could have been produced by one of two plain-text blocks. So if we limit ourselves to reversible mappings, the number of different transformations is 2n!.2

Reversible Mapping Irreversible Mapping

Plaintext Ciphertext Plaintext Ciphertext

00 11 00 11

01 10 01 10

10 00 10 01

11 01 11 01


Figure 3.2 illustrates the logic of a general substitution cipher for n = 4. A 4-bit input produces one of 16 possible input states, which is mapped by the sub-stitution cipher into a unique one of 16 possible output states, each of which is repre-sented by 4 ciphertext bits. The encryption and decryption mappings can be defined by a tabulation, as shown in Table 3.1. This is the most general form of block cipher and can be used to define any reversible mapping between plaintext and ciphertext.

4-bit input

4 to 16 decoder

16 to 4 encoder

4-bit output

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Figure 3.2 General n-bit-n-bit Block Substitution (shown with n = 4)

Table 3.1 Encryption and Decryption Tables for Substitution Cipher of Figure 3.2

Plaintext Ciphertext

0000 11100001 01000010 11010011 00010100 00100101 11110110 10110111 10001000 00111001 10101010 01101011 11001100 01011101 10011110 00001111 0111

Ciphertext Plaintext

0000 11100001 00110010 01000011 10000100 00010101 11000110 10100111 11111000 01111001 11011010 10011011 01101100 10111101 00101110 00001111 0101


Feistel refers to this as the ideal block cipher, because it allows for the maximum number of possible encryption mappings from the plaintext block [FEIS75].

But there is a practical problem with the ideal block cipher. If a small block size, such as n = 4, is used, then the system is equivalent to a classical substitution cipher. Such systems, as we have seen, are vulnerable to a statistical analysis of the plaintext. This weakness is not inherent in the use of a substitution cipher but rather results from the use of a small block size. If n is sufficiently large and an arbitrary re-versible substitution between plaintext and ciphertext is allowed, then the statistical characteristics of the source plaintext are masked to such an extent that this type of cryptanalysis is infeasible.

An arbitrary reversible substitution cipher (the ideal block cipher) for a large block size is not practical, however, from an implementation and performance point of view. For such a transformation, the mapping itself constitutes the key. Consider again Table 3.1, which defines one particular reversible mapping from plaintext to ciphertext for n = 4. The mapping can be defined by the entries in the second column, which show the value of the ciphertext for each plaintext block. This, in essence, is the key that determines the specific mapping from among all pos-sible mappings. In this case, using this straightforward method of defining the key, the required key length is (4 bits) * (16 rows) = 64 bits. In general, for an n-bit ideal block cipher, the length of the key defined in this fashion is n * 2n bits. For a 64-bit block, which is a desirable length to thwart statistical attacks, the required key length is 64 * 264 = 270 ≈ 1021 bits.

In considering these difficulties, Feistel points out that what is needed is an approximation to the ideal block cipher system for large n, built up out of compo-nents that are easily realizable [FEIS75]. But before turning to Feistel’s approach, let us make one other observation. We could use the general block substitution cipher but, to make its implementation tractable, confine ourselves to a subset of the 2n! possible reversible mappings. For example, suppose we define the mapping in terms of a set of linear equations. In the case of n = 4, we have

y1 = k11x1 + k12x2 + k13x3 + k14x4

y2 = k21x1 + k22x2 + k23x3 + k24x4

y3 = k31x1 + k32x2 + k33x3 + k34x4

y4 = k41x1 + k42x2 + k43x3 + k44x4

where the xi are the four binary digits of the plaintext block, the yi are the four binary digits of the ciphertext block, the kij are the binary coefficients, and arithme-tic is mod 2. The key size is just n2, in this case 16 bits. The danger with this kind of formulation is that it may be vulnerable to cryptanalysis by an attacker that is aware of the structure of the algorithm. In this example, what we have is essentially the Hill cipher discussed in Chapter 2, applied to binary data rather than characters. As we saw in Chapter 2, a simple linear system such as this is quite vulnerable.

The Feistel Cipher

Feistel proposed [FEIS73] that we can approximate the ideal block cipher by utilizing the concept of a product cipher, which is the execution of two or more simple ciphers in sequence in such a way that the final result or product is cryptographically stronger


than any of the component ciphers. The essence of the approach is to develop a block cipher with a key length of k bits and a block length of n bits, allowing a total of 2k possible transformations, rather than the 2n! transformations available with the ideal block cipher.

In particular, Feistel proposed the use of a cipher that alternates substitutions and permutations, where these terms are defined as follows:

• Substitution: Each plaintext element or group of elements is uniquely replaced by a corresponding ciphertext element or group of elements.

• Permutation: A sequence of plaintext elements is replaced by a permutation of that sequence. That is, no elements are added or deleted or replaced in the sequence, rather the order in which the elements appear in the sequence is changed.

In fact, Feistel’s is a practical application of a proposal by Claude Shannon to develop a product cipher that alternates confusion and diffusion functions [SHAN49].3 We look next at these concepts of diffusion and confusion and then present the Feistel cipher. But first, it is worth commenting on this remarkable fact: The Feistel cipher structure, which dates back over a quarter century and which, in turn, is based on Shannon’s proposal of 1945, is the structure used by many signifi-cant symmetric block ciphers currently in use.

Diffusion anD Confusion The terms diffusion and confusion were introduced by Claude Shannon to capture the two basic building blocks for any cryptographic system [SHAN49]. Shannon’s concern was to thwart cryptanalysis based on statisti-cal analysis. The reasoning is as follows. Assume the attacker has some knowledge of the statistical characteristics of the plaintext. For example, in a human-readable message in some language, the frequency distribution of the various letters may be known. Or there may be words or phrases likely to appear in the message (probable words). If these statistics are in any way reflected in the ciphertext, the cryptanalyst may be able to deduce the encryption key, part of the key, or at least a set of keys likely to contain the exact key. In what Shannon refers to as a strongly ideal cipher, all statistics of the ciphertext are independent of the particular key used. The arbi-trary substitution cipher that we discussed previously (Figure 3.2) is such a cipher, but as we have seen, it is impractical.4

Other than recourse to ideal systems, Shannon suggests two methods for frustrating statistical cryptanalysis: diffusion and confusion. In diffusion, the statistical structure of the plaintext is dissipated into long-range statistics of the ciphertext. This is achieved by having each plaintext digit affect the value of many

3The paper is available at this book’s Premium Content Web site. Shannon’s 1949 paper appeared origi-nally as a classified report in 1945. Shannon enjoys an amazing and unique position in the history of computer and information science. He not only developed the seminal ideas of modern cryptography but is also responsible for inventing the discipline of information theory. Based on his work in information theory, he developed a formula for the capacity of a data communications channel, which is still used today. In addition, he founded another discipline, the application of Boolean algebra to the study of digi-tal circuits; this last he managed to toss off as a master’s thesis.4Appendix F expands on Shannon’s concepts concerning measures of secrecy and the security of crypto-graphic algorithms.


ciphertext digits; generally, this is equivalent to having each ciphertext digit be affected by many plaintext digits. An example of diffusion is to encrypt a message M = m1, m2, m3, . . . of characters with an averaging operation:

yn = aak

i = 1mn + ib mod 26

adding k successive letters to get a ciphertext letter yn. One can show that the sta-tistical structure of the plaintext has been dissipated. Thus, the letter frequencies in the ciphertext will be more nearly equal than in the plaintext; the digram frequen-cies will also be more nearly equal, and so on. In a binary block cipher, diffusion can be achieved by repeatedly performing some permutation on the data followed by applying a function to that permutation; the effect is that bits from different posi-tions in the original plaintext contribute to a single bit of ciphertext.5

Every block cipher involves a transformation of a block of plaintext into a block of ciphertext, where the transformation depends on the key. The mechanism of diffusion seeks to make the statistical relationship between the plaintext and ciphertext as complex as possible in order to thwart attempts to deduce the key. On the other hand, confusion seeks to make the relationship between the statistics of the ciphertext and the value of the encryption key as complex as possible, again to thwart attempts to discover the key. Thus, even if the attacker can get some handle on the statistics of the ciphertext, the way in which the key was used to produce that ciphertext is so complex as to make it difficult to deduce the key. This is achieved by the use of a complex substitution algorithm. In contrast, a simple linear substitution function would add little confusion.

As [ROBS95b] points out, so successful are diffusion and confusion in captur-ing the essence of the desired attributes of a block cipher that they have become the cornerstone of modern block cipher design.

feistel Cipher struCture The left-hand side of Figure 3.3 depicts the structure proposed by Feistel. The inputs to the encryption algorithm are a plaintext block of length 2w bits and a key K. The plaintext block is divided into two halves, L0 and R0. The two halves of the data pass through n rounds of processing and then combine to produce the ciphertext block. Each round i has as inputs Li- 1 and Ri- 1 derived from the previous round, as well as a subkey Ki derived from the overall K. In general, the subkeys Ki are different from K and from each other. In Figure 3.3, 16 rounds are used, although any number of rounds could be implemented.

All rounds have the same structure. A substitution is performed on the left half of the data. This is done by applying a round function F to the right half of the data and then taking the exclusive-OR of the output of that function and the left half of the data. The round function has the same general structure for each round but is param-eterized by the round subkey Ki. Another way to express this is to say that F is a func-tion of right-half block of w bits and a subkey of y bits, which produces an output value

5Some books on cryptography equate permutation with diffusion. This is incorrect. Permutation, by itself, does not change the statistics of the plaintext at the level of individual letters or permuted blocks. For example, in DES, the permutation swaps two 32-bit blocks, so statistics of strings of 32 bits or less are preserved.


of length w bits: F(REi, Ki+ 1). Following this substitution, a permutation is performed that consists of the interchange of the two halves of the data.6 This structure is a par-ticular form of the substitution-permutation network (SPN) proposed by Shannon.

Output (ciphertext)

K1

LD0 = RE16 RD0 = LE16

LD2 = RE14 RD2 = LE14

LD14 = RE2 RD14 = LE2

LD16 = RE0

LD17 = RE0

RD16 = LE0

RD17 = LE0

RD1 = LE15LD1 = RE15

RD15 = LE1LD15 = RE1

Input (ciphertext)

Output (plaintext)

Rou

nd 1

K1

K2

K15

K16

K2

K15

K16

F

LE0 RE0

Input (plaintext)

LE1 RE1

LE2 RE2

F

F

LE14 RE14

LE15 RE15

LE16 RE16

LE17 RE17

F

F

F

F

F

Rou

nd 2

Rou

nd 1

5R

ound

16

Rou

nd 1

6R

ound

15

Rou

nd 2

Rou

nd 1

Figure 3.3 Feistel Encryption and Decryption (16 rounds)

6The final round is followed by an interchange that undoes the interchange that is part of the final round. One could simply leave both interchanges out of the diagram, at the sacrifice of some consistency of pre-sentation. In any case, the effective lack of a swap in the final round is done to simplify the implementa-tion of the decryption process, as we shall see.


The exact realization of a Feistel network depends on the choice of the follow-ing parameters and design features:

• Block size: Larger block sizes mean greater security (all other things being equal) but reduced encryption/decryption speed for a given algorithm. The greater security is achieved by greater diffusion. Traditionally, a block size of 64 bits has been considered a reasonable tradeoff and was nearly universal in block cipher design. However, the new AES uses a 128-bit block size.

• Key size: Larger key size means greater security but may decrease encryption/decryption speed. The greater security is achieved by greater resistance to brute-force attacks and greater confusion. Key sizes of 64 bits or less are now widely considered to be inadequate, and 128 bits has become a common size.

• Number of rounds: The essence of the Feistel cipher is that a single round offers inadequate security but that multiple rounds offer increasing security. A typical size is 16 rounds.

• Subkey generation algorithm: Greater complexity in this algorithm should lead to greater difficulty of cryptanalysis.

• Round function F: Again, greater complexity generally means greater resis-tance to cryptanalysis.

There are two other considerations in the design of a Feistel cipher:

• Fast software encryption/decryption: In many cases, encryption is embedded in applications or utility functions in such a way as to preclude a hardware imple-mentation. Accordingly, the speed of execution of the algorithm becomes a concern.

• Ease of analysis: Although we would like to make our algorithm as difficult as possible to cryptanalyze, there is great benefit in making the algorithm easy to analyze. That is, if the algorithm can be concisely and clearly explained, it is easier to analyze that algorithm for cryptanalytic vulnerabilities and therefore develop a higher level of assurance as to its strength. DES, for example, does not have an easily analyzed functionality.

feistel DeCryption algorithm The process of decryption with a Feistel cipher is essentially the same as the encryption process. The rule is as follows: Use the ciphertext as input to the algorithm, but use the subkeys Ki in reverse order. That is, use Kn in the first round, Kn - 1 in the second round, and so on, until K1 is used in the last round. This is a nice feature, because it means we need not implement two different algorithms; one for encryption and one for decryption.

To see that the same algorithm with a reversed key order produces the correct result, Figure 3.3 shows the encryption process going down the left-hand side and the decryption process going up the right-hand side for a 16-round algorithm. For clarity, we use the notation LEi and REi for data traveling through the encryption algorithm and LDi and RDi for data traveling through the decryption algorithm. The diagram indicates that, at every round, the intermediate value of the decryption process is equal to the corresponding value of the encryption process with the two halves of the value swapped. To put this another way, let the output of the ith encryption round be


LEi 7REi (LEi concatenated with REi). Then the corresponding output of the (16 – i)th decryption round is REi 7LEi or, equivalently, LD16 - i 7RD16 - i.

Let us walk through Figure 3.3 to demonstrate the validity of the preceding assertions. After the last iteration of the encryption process, the two halves of the output are swapped, so that the ciphertext is RE16 7LE16. The output of that round is the ciphertext. Now take that ciphertext and use it as input to the same algorithm. The input to the first round is RE16 7LE16, which is equal to the 32-bit swap of the output of the sixteenth round of the encryption process.

Now we would like to show that the output of the first round of the decryption process is equal to a 32-bit swap of the input to the sixteenth round of the encryption process. First, consider the encryption process. We see that

LE16 = RE15

RE16 = LE15 ⊕ F(RE15, K16)

On the decryption side,

LD1 = RD0 = LE16 = RE15

RD1 = LD0 ⊕ F(RD0, K16) = RE16 ⊕ F(RE15, K16) = [LE15 ⊕ F(RE15, K16)] ⊕ F(RE15, K16)

The XOR has the following properties:

[A ⊕ B] ⊕ C = A ⊕ [B ⊕ C] D ⊕ D = 0 E ⊕ 0 = E

Thus, we have LD1 = RE15 and RD1 = LE15. Therefore, the output of the first round of the decryption process is RE15 7LE15, which is the 32-bit swap of the input to the sixteenth round of the encryption. This correspondence holds all the way through the 16 iterations, as is easily shown. We can cast this process in general terms. For the ith iteration of the encryption algorithm,

LEi = REi- 1

REi = LEi- 1 ⊕ F(REi- 1, Ki)

Rearranging terms:

REi- 1 = LEi

LEi- 1 = REi ⊕ F(REi- 1, Ki) = REi ⊕ F(LEi, Ki)

Thus, we have described the inputs to the ith iteration as a function of the outputs, and these equations confirm the assignments shown in the right-hand side of Figure 3.3.

Finally, we see that the output of the last round of the decryption process is RE0 7LE0. A 32-bit swap recovers the original plaintext, demonstrating the validity of the Feistel decryption process.

Note that the derivation does not require that F be a reversible function. To see this, take a limiting case in which F produces a constant output (e.g., all ones) regardless of the values of its two arguments. The equations still hold.


To help clarify the preceding concepts, let us look at a specific example (Figure 3.4 and focus on the fifteenth round of encryption, corresponding to the sec-ond round of decryption. Suppose that the blocks at each stage are 32 bits (two 16-bit halves) and that the key size is 24 bits. Suppose that at the end of encryption round fourteen, the value of the intermediate block (in hexadecimal) is DE7F03A6. Then LE14 = DE7F and RE14 = 03A6. Also assume that the value of K15 is 12DE52. After round 15, we have LE15 = 03A6 and RE15 = F(03A6, 12DE52) ⊕ DE7F.

Now let’s look at the decryption. We assume that LD1 = RE15 and RD1 = LE15, as shown in Figure 3.3, and we want to demonstrate that LD2 = RE14 and RD2 = LE14. So, we start with LD1 = F(03A6, 12DE52) ⊕ DE7F and RD1 = 03A6. Then, from Figure 3.3, LD2 = 03A6 = RE14 and RD2 = F(03A6, 12DE52) ⊕ [F(03A6, 12DE52) ⊕ DE7F] = DE7F = LE14.

3.2 the data encryptiOn standard

Until the introduction of the Advanced Encryption Standard (AES) in 2001, the Data Encryption Standard (DES) was the most widely used encryption scheme. DES was issued in 1977 by the National Bureau of Standards, now the National Institute of Standards and Technology (NIST), as Federal Information Processing Standard 46 (FIPS PUB 46). The algorithm itself is referred to as the Data Encryption Algorithm (DEA).7 For DEA, data are encrypted in 64-bit blocks using a 56-bit key. The algorithm transforms 64-bit input in a series of steps into a 64-bit output. The same steps, with the same key, are used to reverse the encryption.

Over the years, DES became the dominant symmetric encryption algorithm, especially in financial applications. In 1994, NIST reaffirmed DES for federal use for another five years; NIST recommended the use of DES for applications other

12DE52

12DE52

F

DE7F 03A6

dnuor noitpyrceDdnuor noitpyrcnE

03A6

6A306A30 F(03A6, 12DE52) DE7F F(03A6, 12DE52) DE7F

F(03A6, 12DE52) [F(03A6, 12DE52) DE7F]

= DE7F

FRou

nd 1

5

Rou

nd 2

Figure 3.4 Feistel Example

7The terminology is a bit confusing. Until recently, the terms DES and DEA could be used interchange-ably. However, the most recent edition of the DES document includes a specification of the DEA described here plus the triple DEA (TDEA) described in Chapter 6. Both DEA and TDEA are part of the Data Encryption Standard. Further, until the recent adoption of the official term TDEA, the triple DEA algorithm was typically referred to as triple DES and written as 3DES. For the sake of convenience, we will use the term 3DES.

3.2 / the data enCryption standard 73

than the protection of classified information. In 1999, NIST issued a new version of its standard (FIPS PUB 46-3) that indicated that DES should be used only for legacy systems and that triple DES (which in essence involves repeating the DES algorithm three times on the plaintext using two or three different keys to produce the ciphertext) be used. We study triple DES in Chapter 6. Because the underly-ing encryption and decryption algorithms are the same for DES and triple DES, it remains important to understand the DES cipher. This section provides an over-view. For the interested reader, Appendix S provides further detail.

DES Encryption

The overall scheme for DES encryption is illustrated in Figure 3.5. As with any en-cryption scheme, there are two inputs to the encryption function: the plaintext to be

Initial permutation

Permuted choice 2Round 1

32-bit swap

Inverse initialpermutation

Permuted choice 1

Round 2

Round 16

64-bit plaintext 64-bit key

K1

K2

K16

64-bit ciphertext

Left circular shift

Permuted choice 2 Left circular shift

Permuted choice 2 Left circular shift

64 56

56

56

56

48

48

48

56 64

64 bits

• • • • • • • • • • • • • • • • • •

• • • • • • • • •

Figure 3.5 General Depiction of DES Encryption Algorithm


encrypted and the key. In this case, the plaintext must be 64 bits in length and the key is 56 bits in length.8

Looking at the left-hand side of the figure, we can see that the processing of the plaintext proceeds in three phases. First, the 64-bit plaintext passes through an initial permutation (IP) that rearranges the bits to produce the permuted input. This is followed by a phase consisting of sixteen rounds of the same function, which involves both permutation and substitution functions. The output of the last (six-teenth) round consists of 64 bits that are a function of the input plaintext and the key. The left and right halves of the output are swapped to produce the preoutput. Finally, the preoutput is passed through a permutation [IP -1] that is the inverse of the initial permutation function, to produce the 64-bit ciphertext. With the excep-tion of the initial and final permutations, DES has the exact structure of a Feistel cipher, as shown in Figure 3.3.

The right-hand portion of Figure 3.5 shows the way in which the 56-bit key is used. Initially, the key is passed through a permutation function. Then, for each of the sixteen rounds, a subkey (Ki) is produced by the combination of a left circular shift and a permutation. The permutation function is the same for each round, but a different subkey is produced because of the repeated shifts of the key bits.

DES Decryption

As with any Feistel cipher, decryption uses the same algorithm as encryption, except that the application of the subkeys is reversed. Additionally, the initial and final permutations are reversed.

3.3 a des exampLe

We now work through an example and consider some of its implications. Although you are not expected to duplicate the example by hand, you will find it informative to study the hex patterns that occur from one step to the next.

For this example, the plaintext is a hexadecimal palindrome. The plaintext, key, and resulting ciphertext are as follows:

Plaintext: 02468aceeca86420

Key: 0f1571c947d9e859

Ciphertext: da02ce3a89ecac3b

Results

Table 3.2 shows the progression of the algorithm. The first row shows the 32-bit values of the left and right halves of data after the initial permutation. The next 16 rows show the results after each round. Also shown is the value of the 48-bit subkey

8Actually, the function expects a 64-bit key as input. However, only 56 of these bits are ever used; the other 8 bits can be used as parity bits or simply set arbitrarily.

3.3 / a des example 75

generated for each round. Note that Li = Ri- 1. The final row shows the left- and right-hand values after the inverse initial permutation. These two values combined form the ciphertext.

The Avalanche Effect

A desirable property of any encryption algorithm is that a small change in either the plaintext or the key should produce a significant change in the ciphertext. In particular, a change in one bit of the plaintext or one bit of the key should produce a change in many bits of the ciphertext. This is referred to as the avalanche effect. If the change were small, this might provide a way to reduce the size of the plaintext or key space to be searched.

Using the example from Table 3.2, Table 3.3 shows the result when the fourth bit of the plaintext is changed, so that the plaintext is 12468aceeca86420. The second column of the table shows the intermediate 64-bit values at the end of each round for the two plaintexts. The third column shows the number of bits that differ between the two intermediate values. The table shows that, after just three rounds, 18 bits differ between the two blocks. On completion, the two ciphertexts differ in 32 bit positions.

Table 3.4 shows a similar test using the original plaintext of with two keys that differ in only the fourth bit position: the original key, 0f1571c947d9e859, and the altered key, 1f1571c947d9e859. Again, the results show that about half of the bits in the ciphertext differ and that the avalanche effect is pronounced after just a few rounds.

Table 3.2 DES Example

Round Ki Li Ri

IP 5a005a00 3cf03c0f

1 1e030f03080d2930 3cf03c0f bad22845

2 0a31293432242318 bad22845 99e9b723

3 23072318201d0c1d 99e9b723 0bae3b9e

4 05261d3824311a20 0bae3b9e 42415649

5 3325340136002c25 42415649 18b3fa41

6 123a2d0d04262a1c 18b3fa41 9616fe23

7 021f120b1c130611 9616fe23 67117cf2

8 1c10372a2832002b 67117cf2 c11bfc09

9 04292a380c341f03 c11bfc09 887fbc6c

10 2703212607280403 887fbc6c 600f7e8b

11 2826390c31261504 600f7e8b f596506e

12 12071c241a0a0f08 f596506e 738538b8

13 300935393c0d100b 738538b8 c6a62c4e

14 311e09231321182a c6a62c4e 56b0bd75

15 283d3e0227072528 56b0bd75 75e8fd8f

16 2921080b13143025 75e8fd8f 25896490

IP–1 da02ce3a 89ecac3b

Note: DES subkeys are shown as eight 6-bit values in hex format


Table 3.3 Avalanche Effect in DES: Change in Plaintext

Round 802468aceeca86420 12468aceeca86420

1

1 3cf03c0fbad22845 3cf03c0fbad32845

1

2 bad2284599e9b723 bad3284539a9b7a3

5

3 99e9b7230bae3b9e 39a9b7a3171cb8b3

18

4 0bae3b9e42415649 171cb8b3ccaca55e

34

5 4241564918b3fa41 ccaca55ed16c3653

37

6 18b3fa419616fe23 d16c3653cf402c68

33

7 9616fe2367117cf2 cf402c682b2cefbc

32

8 67117cf2c11bfc09 2b2cefbc99f91153

33

Round 89 c11bfc09887fbc6c

99f911532eed7d9432

10 887fbc6c600f7e8b 2eed7d94d0f23094

34

11 600f7e8bf596506e d0f23094455da9c4

37

12 f596506e738538b8 455da9c47f6e3cf3

31

13 738538b8c6a62c4e 7f6e3cf34bc1a8d9

29

14 c6a62c4e56b0bd75 4bc1a8d91e07d409

33

15 56b0bd7575e8fd8f 1e07d4091ce2e6dc

31

16 75e8fd8f25896490 1ce2e6dc365e5f59

32

IP–1 da02ce3a89ecac3b 057cde97d7683f2a

32

Table 3.4 Avalanche Effect in DES: Change in Key

Round 802468aceeca86420 02468aceeca86420

0

1 3cf03c0fbad22845 3cf03c0f9ad628c5

3

2 bad2284599e9b723 9ad628c59939136b

11

3 99e9b7230bae3b9e 9939136b768067b7

25

4 0bae3b9e42415649 768067b75a8807c5

29

5 4241564918b3fa41 5a8807c5488dbe94

26

6 18b3fa419616fe23 488dbe94aba7fe53

26

7 9616fe2367117cf2 aba7fe53177d21e4

27

8 67117cf2c11bfc09 177d21e4548f1de4

32

Round 89 c11bfc09887fbc6c

548f1de471f64dfd34

10 887fbc6c600f7e8b 71f64dfd4279876c

36

11 600f7e8bf596506e 4279876c399fdc0d

32

12 f596506e738538b8 399fdc0d6d208dbb

28

13 738538b8c6a62c4e 6d208dbbb9bdeeaa

33

14 c6a62c4e56b0bd75 b9bdeeaad2c3a56f

30

15 56b0bd7575e8fd8f d2c3a56f2765c1fb

33

16 75e8fd8f25896490 2765c1fb01263dc4

30

IP–1 da02ce3a89ecac3b ee92b50606b62b0b

30

3.4 / the strength of des 77

3.4 the strength Of des

Since its adoption as a federal standard, there have been lingering concerns about the level of security provided by DES. These concerns, by and large, fall into two areas: key size and the nature of the algorithm.

The Use of 56-Bit Keys

With a key length of 56 bits, there are 256 possible keys, which is approximately 7.2 * 1016 keys. Thus, on the face of it, a brute-force attack appears impractical. Assuming that, on average, half the key space has to be searched, a single machine performing one DES encryption per microsecond would take more than a thousand years to break the cipher.

However, the assumption of one encryption per microsecond is overly con-servative. As far back as 1977, Diffie and Hellman postulated that the technology existed to build a parallel machine with 1 million encryption devices, each of which could perform one encryption per microsecond [DIFF77]. This would bring the average search time down to about 10 hours. The authors estimated that the cost would be about $20 million in 1977 dollars.

With current technology, it is not even necessary to use special, purpose-built hardware. Rather, the speed of commercial, off-the-shelf processors threaten the security of DES. A recent paper from Seagate Technology [SEAG08] suggests that a rate of 1 billion (109) key combinations per second is reasonable for today’s mul-ticore computers. Recent offerings confirm this. Both Intel and AMD now offer hardware-based instructions to accelerate the use of AES. Tests run on a contem-porary multicore Intel machine resulted in an encryption rate of about half a bil-lion encryptions per second [BASU12]. Another recent analysis suggests that with contemporary supercomputer technology, a rate of 1013 encryptions per second is reasonable [AROR12].

With these results in mind, Table 3.5 shows how much time is required for a brute-force attack for various key sizes. As can be seen, a single PC can break DES in about a year; if multiple PCs work in parallel, the time is drastically short-ened. And today’s supercomputers should be able to find a key in about an hour. Key sizes of 128 bits or greater are effectively unbreakable using simply a brute-force approach. Even if we managed to speed up the attacking system by a factor of 1 trillion (1012), it would still take over 100,000 years to break a code using a 128-bit key.

Fortunately, there are a number of alternatives to DES, the most important of which are AES and triple DES, discussed in Chapters 5 and 6, respectively.

The Nature of the DES Algorithm

Another concern is the possibility that cryptanalysis is possible by exploiting the characteristics of the DES algorithm. The focus of concern has been on the eight substitution tables, or S-boxes, that are used in each iteration (described in Appendix S). Because the design criteria for these boxes, and indeed for the entire algorithm, were not made public, there is a suspicion that the boxes were


constructed in such a way that cryptanalysis is possible for an opponent who knows the weaknesses in the S-boxes. This assertion is tantalizing, and over the years a number of regularities and unexpected behaviors of the S-boxes have been discov-ered. Despite this, no one has so far succeeded in discovering the supposed fatal weaknesses in the S-boxes.9

Timing Attacks

We discuss timing attacks in more detail in Part Two, as they relate to public-key algorithms. However, the issue may also be relevant for symmetric ciphers. In essence, a timing attack is one in which information about the key or the plaintext is obtained by observing how long it takes a given implementation to perform decryptions on various ciphertexts. A timing attack exploits the fact that an encryption or decryption algorithm often takes slightly different amounts of time on different inputs. [HEVI99] reports on an approach that yields the Hamming weight (number of bits equal to one) of the secret key. This is a long way from knowing the actual key, but it is an intriguing first step. The authors conclude that DES appears to be fairly resistant to a successful timing attack but suggest some avenues to explore. Although this is an interesting line of attack, it so far appears unlikely that this technique will ever be successful against DES or more powerful symmetric ciphers such as triple DES and AES.

3.5 bLOck cipher design principLes

Although much progress has been made in designing block ciphers that are cryp-tographically strong, the basic principles have not changed all that much since the work of Feistel and the DES design team in the early 1970s. In this section we look at three critical aspects of block cipher design: the number of rounds, design of the function F, and key scheduling.

Table 3.5 Average Time Required for Exhaustive Key Search

Key Size (bits) CipherNumber of

Alternative KeysTime Required at 109

Decryptions/s

Time Required at 1013

Decryptions/s

56 DES 256 L 7.2 * 1016 255 ns = 1.125 years 1 hour

128 AES 2128 L 3.4 * 1038 2127 ns = 5.3 * 1021 years 5.3 * 1017 years

168 Triple DES 2168 L 3.7 * 1050 2167 ns = 5.8 * 1033 years 5.8 * 1029 years



26 characters (permutation)

Monoalphabetic 2! = 4 * 1026 2 * 1026 ns = 6.3 * 109 years 6.3 * 106 years

9At least, no one has publicly acknowledged such a discovery.

3.5 / BloCk Cipher design prinCiples 79

Number of Rounds

The cryptographic strength of a Feistel cipher derives from three aspects of the design: the number of rounds, the function F, and the key schedule algorithm. Let us look first at the choice of the number of rounds.

The greater the number of rounds, the more difficult it is to perform crypt-analysis, even for a relatively weak F. In general, the criterion should be that the number of rounds is chosen so that known cryptanalytic efforts require greater effort than a simple brute-force key search attack. This criterion was certainly used in the design of DES. Schneier [SCHN96] observes that for 16-round DES, a differ-ential cryptanalysis attack is slightly less efficient than brute force: The differential cryptanalysis attack requires 255.1 operations,10 whereas brute force requires 255. If DES had 15 or fewer rounds, differential cryptanalysis would require less effort than a brute-force key search.

This criterion is attractive, because it makes it easy to judge the strength of an algorithm and to compare different algorithms. In the absence of a cryptana-lytic breakthrough, the strength of any algorithm that satisfies the criterion can be judged solely on key length.

Design of Function F

The heart of a Feistel block cipher is the function F, which provides the element of confusion in a Feistel cipher. Thus, it must be difficult to “unscramble” the substitution performed by F. One obvious criterion is that F be nonlinear, as we discussed previously. The more nonlinear F, the more difficult any type of crypt-analysis will be. There are several measures of nonlinearity, which are beyond the scope of this book. In rough terms, the more difficult it is to approximate F by a set of linear equations, the more nonlinear F is.

Several other criteria should be considered in designing F. We would like the algorithm to have good avalanche properties. Recall that, in general, this means that a change in one bit of the input should produce a change in many bits of the output. A more stringent version of this is the strict avalanche criterion (SAC) [WEBS86], which states that any output bit j of an S-box (see Appendix S for a discussion of S-boxes) should change with probability 1/2 when any single input bit i is inverted for all i, j. Although SAC is expressed in terms of S-boxes, a similar criterion could be applied to F as a whole. This is important when considering designs that do not include S-boxes.

Another criterion proposed in [WEBS86] is the bit independence criterion (BIC), which states that output bits j and k should change independently when any single input bit i is inverted for all i, j, and k. The SAC and BIC criteria appear to strengthen the effectiveness of the confusion function.

10Differential cryptanalysis of DES requires 247 chosen plaintext. If all you have to work with is known plaintext, then you must sort through a large quantity of known plaintext–ciphertext pairs looking for the useful ones. This brings the level of effort up to 255.1.


Key Schedule Algorithm

With any Feistel block cipher, the key is used to generate one subkey for each round. In general, we would like to select subkeys to maximize the difficulty of deducing individual subkeys and the difficulty of working back to the main key. No general principles for this have yet been promulgated.

Adams suggests [ADAM94] that, at minimum, the key schedule should guarantee key/ciphertext Strict Avalanche Criterion and Bit Independence Criterion.


There is a wealth of information on symmetric encryption. Some of the more worthwhile references are listed here. An essential reference work is [SCHN96]. This remarkable work contains descriptions of virtually every cryptographic algorithm and protocol pub-lished up to the time of the writing of the book. The author pulls together results from journals, conference proceedings, government publications, and standards documents and organizes these into a comprehensive and comprehensible survey. Another worthwhile and detailed survey is [MENE97]. A rigorous mathematical treatment is [STIN06].

The foregoing references provide coverage of public-key as well as symmetric encryption.

Perhaps the most detailed description of DES is [SIMO95]; the book also con-tains an extensive discussion of differential and linear cryptanalysis of DES. [BARK91] provides a readable and interesting analysis of the structure of DES and of potential cryptanalytic approaches to DES. [EFF98] details the most effective brute-force attack on DES. [COPP94] looks at the inherent strength of DES and its ability to stand up to cryptanalysis. The reader may also find the following document useful: “The DES Algorithm Illustrated” by J. Orlin Grabbe, which is available at this book’s Premium Content Web site.

BARK91 Barker, W. Introduction to the Analysis of the Data Encryption Standard (DES). Laguna Hills, CA: Aegean Park Press, 1991.

COPP94 Coppersmith, D. “The Data Encryption Standard (DES) and Its Strength Against Attacks.” IBM Journal of Research and Development, May 1994.

EFF98 Electronic Frontier Foundation. Cracking DES: Secrets of Encryption Research, Wiretap Politics, and Chip Design. Sebastopol, CA: O’Reilly, 1998.

MENE97 Menezes, A., van Oorschot, P., and Vanstone, S. Handbook of Applied Cryptography. Boca Raton, FL: CRC Press, 1997.

SCHN96 Schneier, B. Applied Cryptography. New York: Wiley, 1996.SIMO95 Simovits, M. The DES: An Extensive Documentation and Evaluation. Laguna

Hills, CA: Aegean Park Press, 1995.STIN06 Stinson, D. Cryptography: Theory and Practice. Boca Raton, FL: Chapman &

Hall, 2006.

3.7 / key terms, review Questions, and proBlems 81

3.7 key terms, review QuestiOns, and prObLems

Key Terms

avalanche effectblock cipherconfusionData Encryption Standard

(DES)diffusion

Feistel cipherirreversible mappingkeypermutationproduct cipher reversible mapping

roundround functionsubkeysubstitution

Review Questions 3.1 Why is it important to study the Feistel cipher? 3.2 What is the difference between a block cipher and a stream cipher? 3.3 Why is it not practical to use an arbitrary reversible substitution cipher of the kind

shown in Table 3.1? 3.4 What is a product cipher? 3.5 What is the difference between diffusion and confusion? 3.6 Which parameters and design choices determine the actual algorithm of a Feistel

cipher? 3.7 Explain the avalanche effect.

Problems 3.1 a. In Section 3.1, under the subsection on the motivation for the Feistel cipher struc-

ture, it was stated that, for a block of n bits, the number of different reversible mappings for the ideal block cipher is 2n!. Justify.

b. In that same discussion, it was stated that for the ideal block cipher, which allows all possible reversible mappings, the size of the key is n * 2n bits. But, if there are 2n! possible mappings, it should take log2 2

n! bits to discriminate among the different mappings, and so the key length should be log2 2

n!. However, log2 2n! 6 n * 2n.

Explain the discrepancy. 3.2 Consider a Feistel cipher composed of sixteen rounds with a block length of 128 bits

and a key length of 128 bits. Suppose that, for a given k, the key scheduling algorithm determines values for the first eight round keys, k1, k2, c k8, and then sets

k9 = k8, k10 = k7, k11 = k6, c , k16 = k1

Suppose you have a ciphertext c. Explain how, with access to an encryption oracle, you can decrypt c and determine m using just a single oracle query. This shows that such a cipher is vulnerable to a chosen plaintext attack. (An encryption oracle can be thought of as a device that, when given a plaintext, returns the corresponding cipher-text. The internal details of the device are not known to you and you cannot break open the device. You can only gain information from the oracle by making queries to it and observing its responses.)

3.3 Let p be a permutation of the integers 0, 1, 2, c, (2n - 1), such that p(m) gives the permuted value of m, 0 … m 6 2n. Put another way, p maps the set of n-bit integers into itself and no two integers map into the same integer. DES is such a permutation for 64-bit integers. We say that p has a fixed point at m if p(m) = m. That is, if p is


an encryption mapping, then a fixed point corresponds to a message that encrypts to itself. We are interested in the probability that p has no fixed points. Show the some-what unexpected result that over 60% of mappings will have at least one fixed point.

3.4 Consider a block encryption algorithm that encrypts blocks of length n, and let N = 2n. Say we have t plaintext–ciphertext pairs Pi, Ci = E(K, Pi), where we assume that the key K selects one of the N! possible mappings. Imagine that we wish to find K by exhaustive search. We could generate key K′ and test whether Ci = E(K′, Pi) for 1 … i … t. If K′ encrypts each Pi to its proper Ci, then we have evidence that K = K′. However, it may be the case that the mappings E(K, #) and E(K′, #) exactly agree on the t plaintext–cipher text pairs Pi, Ci and agree on no other pairs.

a. What is the probability that E(K, #) and E(K′, #) are in fact distinct mappings? b. What is the probability that E(K, #) and E(K′, #) agree on another t′ plaintext–

ciphertext pairs where 0 … t′ … N - t? 3.5 For any block cipher, the fact that it is a nonlinear function is crucial to its security. To

see this, suppose that we have a linear block cipher EL that encrypts 128-bit blocks of plaintext into 128-bit blocks of ciphertext. Let EL (k, m) denote the encryption of a 128-bit message m under a key k (the actual bit length of k is irrelevant). Thus,

EL(k, [m1 ⊕ m2]) = EL(k, m1) ⊕ EL(k, m2) for all 128@bit patterns m1, m2

Describe how, with 128 chosen ciphertexts, an adversary can decrypt any ciphertext without knowledge of the secret key k. (A “chosen ciphertext” means that an adver-sary has the ability to choose a ciphertext and then obtain its decryption. Here, you have 128 plaintext/ciphertext pairs to work with and you have the ability to chose the value of the ciphertexts.)

3.6 Suppose the DES F function mapped every 32-bit input R, regardless of the value of the input K, to

a. 32-bit string of ones b. bitwise complement of R

Hint: Use the following properties of the XOR operation: 1. What function would DES then compute? 2. What would the decryption look like?

(A ⊕ B) ⊕ C = A ⊕ (B ⊕ C) A ⊕ A = 0 A ⊕ 0 = A

A ⊕ 1 = bitwise complement of A where

A,B,C are n-bit strings of bits0 is an n-bit string of zeros1 is an n-bit string of one

3.7 Show that DES decryption is, in fact, the inverse of DES encryption. 3.8 The 32-bit swap after the sixteenth iteration of the DES algorithm is needed to make

the encryption process invertible by simply running the ciphertext back through the algorithm with the key order reversed. This was demonstrated in Problem 3.7. However, it still may not be entirely clear why the 32-bit swap is needed. To demon-strate why, solve the following exercises. First, some notation:

A 7B = the concatenation of the bit strings A and BTi(R 7L) = the transformation defined by the ith iteration of the encryption

algorithm for 1 … I … 16TDi(R 7L) = the transformation defined by the ith iteration of the encryption

algorithm for 1 … I … 16

3.7 / key terms, review Questions, and proBlems 83

T17(R 7L) = L 7R, where this transformation occurs after the sixteenth iteration of the encryption algorithm

a. Show that the composition TD1(IP(IP- 1(T17(T16(L15 7R15))))) is equivalent to the transformation that interchanges the 32-bit halves, L15 and R15. That is, show that

TD1(IP(IP-1(T17(T16(L15 7R15))))) = R15 7L15

b. Now suppose that we did away with the final 32-bit swap in the encryption algo-rithm. Then we would want the following equality to hold:

TD1(IP(IP-1(T16(L15 7R15)))) = L15 7R15

Does it?

Note: The following problems refer to details of DES that are described in Appendix S.

3.9 Consider the substitution defined by row 1 of S-box S1 in Table S.2. Show a block diagram similar to Figure 3.2 that corresponds to this substitution.

3.10 Compute the bits number 1, 16, 33, and 48 at the output of the first round of the DES decryption, assuming that the ciphertext block is composed of all ones and the external key is composed of all ones.

3.11 This problem provides a numerical example of encryption using a one-round ver-sion of DES. We start with the same bit pattern for the key K and the plaintext, namely:

Hexadecimal notation: 0 1 2 3 4 5 6 7 8 9 A B C D E F

Binary notation: 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

a. Derive K1, the first-round subkey. b. Derive L0, R0. c. Expand R0 to get E[R0], where E[ #] is the expansion function of Table S.1. d. Calculate A = E[R0] ⊕ K1. e. Group the 48-bit result of (d) into sets of 6 bits and evaluate the corresponding

S-box substitutions. f. Concatenate the results of (e) to get a 32-bit result, B. g. Apply the permutation to get P(B). h. Calculate R1 = P(B) ⊕ L0. i. Write down the ciphertext. 3.12 Compare the initial permutation table (Table S.1a) with the permuted choice one

table (Table S.3b). Are the structures similar? If so, describe the similarities. What conclusions can you draw from this analysis?

3.13 When using the DES algorithm for decryption, the 16 keys (K1, K2, c, K16) are used in reverse order. Therefore, the right-hand side of Figure S.1 is not valid for decryption. Design a key-generation scheme with the appropriate shift schedule (analogous to Table S.3d) for the decryption process.

3.14 a. Let X′ be the bitwise complement of X . Prove that if the complement of the plaintext block is taken and the complement of an encryption key is taken, then the result of DES encryption with these values is the complement of the original ciphertext. That is,

If Y = E(K, X)Then Y′ = E(K′, X′)

Hint: Begin by showing that for any two bit strings of equal length, A and B, (A ⊕ B)′ = A′ ⊕ B.


b. It has been said that a brute-force attack on DES requires searching a key space of 256 keys. Does the result of part (a) change that?

3.15 Show that in DES the first 24 bits of each subkey come from the same subset of 28 bits of the initial key and that the second 24 bits of each subkey come from a disjoint subset of 28 bits of the initial key.

Note: The following problems refer to simplified DES, described in Appendix G.

3.16 Refer to Figure G.2, which depicts key generation for S-DES. a. How important is the initial P10 permutation function? b. How important are the two LS-1 shift functions? 3.17 The equations for the variables q and r for S-DES are defined in the section on

S-DES analysis. Provide the equations for s and t. 3.18 Using S-DES, decrypt the string (10100010) using the key (0111111101) by hand.

Show intermediate results after each function (IP, FK, SW, FK, IP-1). Then decode the first 4 bits of the plaintext string to a letter and the second 4 bits to another letter where we encode A through P in base 2 (i.e., A = 0000, B = 0001, ..., P = 1111). Hint: As a midway check, after the application of SW, the string should be (00010011).

Programming Problems 3.19 Create software that can encrypt and decrypt using a general substitution block

cipher. 3.20 Create software that can encrypt and decrypt using S-DES. Test data: use plaintext,

ciphertext, and key of Problem 3.18.

85

4.1 Divisibility and The Division Algorithm

4.2 The Euclidean Algorithm

4.3 Modular Arithmetic

4.4 Groups, Rings, and Fields

4.5 Finite Fields of the Form GF(p)

4.6 Polynomial Arithmetic

4.7 Finite Fields of the Form GF(2n)



Appendix 4A The Meaning of Mod

Chapter

Basic concepts in numBer theory and Finite Fields

86 Chapter 4 / BasiC ConCepts in numBer theory and Finite Fields

Mathematics has long been known in the printing trade as difficult, or penalty, copy because it is slower, more difficult, and more expensive to set in type than any other kind of copy.

—Chicago Manual of Style, University of Chicago Press, Chicago 60637, © The University of Chicago

Finite fields have become increasingly important in cryptography. A number of cryp-tographic algorithms rely heavily on properties of finite fields, notably the Advanced Encryption Standard (AES) and elliptic curve cryptography. Other examples in-clude the message authentication code CMAC and the authenticated encryption scheme GCM.

This chapter provides the reader with sufficient background on the concepts of finite fields to be able to understand the design of AES and other cryptographic algo-rithms that use finite fields. The first three sections introduce basic concepts from num-ber theory that are needed in the remainder of the chapter; these include divisibility, the Euclidian algorithm, and modular arithmetic. Next comes a brief overview of the concepts of group, ring, and field. This section is somewhat abstract; the reader may prefer to quickly skim this section on a first reading. We are then ready to discuss finite fields of the form GF(p), where p is a prime number. Next, we need some additional background, this time in polynomial arithmetic. The chapter concludes with a discus-sion of finite fields of the form GF(2n), where n is a positive integer.

The concepts and techniques of number theory are quite abstract, and it is often difficult to grasp them intuitively without examples. Accordingly, this chapter and Chapter 8 include a number of examples, each of which is highlighted in a shaded box.

Learning Objectives


u Understand the concept of divisibility and the division algorithm.

u Understand how to use the Euclidean algorithm to find the greatest com-mon divisor.

u Present an overview of the concepts of modular arithmetic.

u Explain the operation of the extended Euclidean algorithm.

u Distinguish among groups, rings, and fields.

u Define finite fields of the form GF(p).

u Explain the differences among ordinary polynomial arithmetic, polynomial arithmetic with coefficients in Zp, and modular polynomial arithmetic in GF(2n).

u Define finite fields of the form GF(2n).

u Explain the two different uses of the mod operator.

4.1 / divisiBility and the division algorithm 87

4.1 DivisibiLity anD the DivisiOn aLgOrithm

Divisibility

We say that a nonzero b divides a if a = mb for some m, where a, b, and m are integers. That is, b divides a if there is no remainder on division. The notation b � a is commonly used to mean b divides a. Also, if b � a, we say that b is a divisor of a.

The positive divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.13 � 182; -5 � 30; 17 � 289; -3 � 33; 17 � 0

b = 7; g = 14; h = 63; m = 3; n = 2

7 � 14 and 7 � 63.

To show 7 � (3 * 14 + 2 * 63),

we have (3 * 14 + 2 * 63) = 7(3 * 2 + 2 * 9),

and it is obvious that 7 � (7(3 * 2 + 2 * 9)).

11 � 66 and 66 � 198 = 11 � 198

Subsequently, we will need some simple properties of divisibility for integers, which are as follows:

• If a � 1, then a = {1.

• If a � b and b � a, then a = {b.

• Any b ≠ 0 divides 0.

• If a � b and b � c, then a � c:

• If b � g and b � h, then b � (mg + nh) for arbitrary integers m and n.

To see this last point, note that

• If b � g, then g is of the form g = b * g1 for some integer g1.

• If b � h, then h is of the form h = b * h1 for some integer h1.

So

mg + nh = mbg1 + nbh1 = b * (mg1 + nh1)

and therefore b divides mg + nh.

The Division Algorithm

Given any positive integer n and any nonnegative integer a, if we divide a by n, we get an integer quotient q and an integer remainder r that obey the following relationship:

a = qn + r 0 … r 6 n; q = :a/n; (4.1)


where :x; is the largest integer less than or equal to x. Equation (4.1) is referred to as the division algorithm.1

Figure 4.1a demonstrates that, given a and positive n, it is always possible to find q and r that satisfy the preceding relationship. Represent the integers on the number line; a will fall somewhere on that line (positive a is shown, a similar dem-onstration can be made for negative a). Starting at 0, proceed to n, 2n, up to qn, such that qn … a and (q + 1)n 7 a. The distance from qn to a is r, and we have found the unique values of q and r. The remainder r is often referred to as a residue.

1Equation (4.1) expresses a theorem rather than an algorithm, but by tradition, this is referred to as the division algorithm.

0

n 2n 3n qn (q + 1)na

n

r(a) General relationship

0 15

15

10

30= 2 15

70

(b) Example: 70 = (4 15) + 10

45= 3 15

60= 4 15

75= 5 15

Figure 4.1 The Relationship a = qn + r; 0 … r 6 n

a = 11; n = 7; 11 = 1 * 7 + 4; r = 4 q = 1a = -11; n = 7; -11 = (-2) * 7 + 3; r = 3 q = -2

Figure 4.1b provides another example.

4.2 the eucLiDean aLgOrithm

One of the basic techniques of number theory is the Euclidean algorithm, which is a simple procedure for determining the greatest common divisor of two positive integers. First, we need a simple definition: Two integers are relatively prime if their only common positive integer factor is 1.

Greatest Common Divisor

Recall that nonzero b is defined to be a divisor of a if a = mb for some m, where a, b, and m are integers. We will use the notation gcd(a, b) to mean the greatest common divisor

4.2 / the euClidean algorithm 89

of a and b. The greatest common divisor of a and b is the largest integer that divides both a and b. We also define gcd(0, 0) = 0.

More formally, the positive integer c is said to be the greatest common divisor of a and b if

1. c is a divisor of a and of b.

2. Any divisor of a and b is a divisor of c.

An equivalent definition is the following:

gcd(a, b) = max[k, such that k � a and k � b]

Because we require that the greatest common divisor be positive, gcd(a, b) =gcd(a, -b) = gcd(-a, b) = gcd(-a,-b). In general, gcd(a, b) = gcd( � a � , � b � ).

gcd(60, 24) = gcd(60, -24) = 12

Also, because all nonzero integers divide 0, we have gcd(a, 0) = � a � .We stated that two integers a and b are relatively prime if their only common

positive integer factor is 1. This is equivalent to saying that a and b are relatively prime if gcd(a, b) = 1.

8 and 15 are relatively prime because the positive divisors of 8 are 1, 2, 4, and 8, and the positive divisors of 15 are 1, 3, 5, and 15. So 1 is the only integer on both lists.

Finding the Greatest Common Divisor

We now describe an algorithm credited to Euclid for easily finding the great-est common divisor of two integers. This algorithm has significance subsequently in this chapter. Suppose we have integers a, b such that d = gcd(a, b). Because gcd( � a � , � b � ) = gcd(a, b), there is no harm in assuming a Ú b 7 0. Now dividing a by b and applying the division algorithm, we can state:

a = q1b + r1 0 … r1 6 b (4.2)

If it happens that r1 = 0, then b � a and d = gcd(a, b) = b. But if r1 ≠ 0, we can state that d � r1. This is due to the basic properties of divisibility: the relations d � a and d � b together imply that d � (a - q1b), which is the same as d � r1. Before proceeding with the Euclidian algorithm, we need to answer the question: What is the gcd(b, r1)? We know that d � b and d � r1. Now take any arbitrary integer c that divides both b and r1. Therefore, c � (q1b + r1) = a. Because c divides both a and b, we must have c … d, which is the greatest common divisor of a and b. Therefore d = gcd(b, r1).

Let us now return to Equation (4.2) and assume that r1 ≠ 0. Because b 7 r1, we can divide b by r1and apply the division algorithm to obtain:

b = q2r1 + r2 0 … r2 6 r1

As before, if r2 = 0, then d = r1 and if r2 ≠ 0, then d = gcd(r1, r2). The divi-sion process continues until some zero remainder appears, say, at the (n + 1)th


stage where rn - 1 is divided by rn. The result is the following system of equations:

a = q1b + r1 0 6 r1 6 bb = q2r1 + r2 0 6 r2 6 r1

r1 = q3r2 + r3 0 6 r3 6 r2

~ ~ ~ ~ ~ ~rn - 2 = qnrn - 1 + rn 0 6 rn 6 rn - 1

rn - 1 = qn + 1rn + 0d = gcd(a, b) = rn

y (4.3)

At each iteration, we have d = gcd(ri, ri+ 1) until finally d = gcd(rn, 0) = rn. Thus, we can find the greatest common divisor of two integers by repetitive applica-tion of the division algorithm. This scheme is known as the Euclidean algorithm.

We have essentially argued from the top down that the final result is the gcd(a, b). We can also argue from the bottom up. The first step is to show that rn divides a and b. It follows from the last division in Equation (4.3) that rn divides rn - 1. The next to last division shows that rn divides rn - 2 because it divides both terms on the right. Successively, one sees that rn divides all ri’s and finally a and b. It remains to show that rn is the largest divisor that divides a and b. If we take any arbi-trary integer that divides a and b, it must also divide r1, as explained previously. We can follow the sequence of equations in Equation (4.3) down and show that c must divide all ri’s. Therefore c must divide rn, so that rn = gcd(a, b).

Let us now look at an example with relatively large numbers to see the power of this algorithm:

To find d = gcd (a,b) = gcd (1160718174, 316258250)

a = q1b + r1 1160718174 = 3 * 316258250 + 211943424 d = gcd(316258250, 211943424)

b = q2r1 + r2 316258250 = 1 * 211943424 + 104314826 d = gcd(211943424, 104314826)

r1 = q3r2 + r3 211943424 = 2 * 104314826 + 3313772 d = gcd(104314826, 3313772)

r2 = q4r3 + r4 104314826 = 31 * 3313772 + 1587894 d = gcd(3313772, 1587894)

r3 = q5r4 + r5 3313772 = 2 * 1587894 + 137984 d = gcd(1587894, 137984)

r4 = q6r5 + r6 1587894 = 11 * 137984 + 70070 d = gcd(137984, 70070)

r5 = q7r6 + r7 137984 = 1 * 70070 + 67914 d = gcd(70070, 67914)

r6 = q8r7 + r8 70070 = 1 * 67914 + 2156 d = gcd(67914, 2156)

r7 = q9r8 + r9 67914 = 31 * 2516 + 1078 d = gcd(2156, 1078)

r8 = q10r9 + r10 2156 = 2 * 1078 + 0 d = gcd(1078, 0) = 1078

Therefore, d = gcd(1160718174, 316258250) = 1078

4.3 / modular arithmetiC 91

In this example, we begin by dividing 1160718174 by 316258250, which gives 3 with a remainder of 211943424. Next we take 316258250 and divide it by 211943424. The process continues until we get a remainder of 0, yielding a result of 1078.

It will be helpful in what follows to recast the above computation in tabular form. For every step of the iteration, we have ri- 2 = qiri- 1 + ri, where ri- 2 is the dividend, ri- 1 is the divisor, qi is the quotient, and ri is the remainder. Table 4.1 sum-marizes the results.

4.3 mODuLar arithmetic

The Modulus

If a is an integer and n is a positive integer, we define a mod n to be the remainder when a is divided by n. The integer n is called the modulus. Thus, for any integer a, we can rewrite Equation (4.1) as follows:

a = qn + r 0 … r 6 n; q = :a/n; a = :a/n; * n + (a mod n)

Table 4.1 Euclidean Algorithm Example

Dividend Divisor Quotient Remainder

a = 1160718174 b = 316258250 q1 = 3 r1 = 211943424

b = 316258250 r1 = 211943434 q2 = 1 r2 = 104314826

r1 = 211943424 r2 = 104314826 q3 = 2 r3 = 3313772

r2 = 104314826 r3 = 3313772 q4 = 31 r4 = 1587894

r3 = 3313772 r4 = 1587894 q5 = 2 r5 = 137984

r4 = 1587894 r5 = 137984 q6 = 11 r6 = 70070

r5 = 137984 r6 = 70070 q7 = 1 r7 = 67914

r6 = 70070 r7 = 67914 q8 = 1 r8 = 2156

r7 = 67914 r8 = 2156 q9 = 31 r9 = 1078

r8 = 2156 r9 = 1078 q10 = 2 r10 = 0

2We have just used the operator mod in two different ways: first as a binary operator that produces a remainder, as in the expression a mod b; second as a congruence relation that shows the equivalence of two integers, as in the expression a K b(mod n). See Appendix 4A for a discussion.

11 mod 7 = 4; - 11 mod 7 = 3

Two integers a and b are said to be congruent modulo n, if (a mod n) =(b mod n). This is written as a K b (mod n).2

73 K 4 (mod 23); 21 K -9 (mod 10)

Note that if a K 0 (mod n), then n � a.


Properties of Congruences

Congruences have the following properties:

1. a K b (mod n) if n � (a - b). 2. a K b (mod n) implies b K a (mod n). 3. a K b (mod n) and b K c (mod n) imply a K c (mod n).

To demonstrate the first point, if n � (a - b), then (a - b) = kn for some k. So we can write a = b + kn. Therefore, (a mod n) = (remainder when b + kn isdivided by n) = (remainder when b is divided by n) = (b mod n).

23 K 8 (mod 5) because 23 - 8 = 15 = 5 * 3-11 K 5 (mod 8) because -11 - 5 = -16 = 8 * (-2)81 K 0 (mod 27) because 81 - 0 = 81 = 27 * 3

The remaining points are as easily proved.

Modular Arithmetic Operations

Note that, by definition (Figure 4.1), the (mod n) operator maps all integers into the set of integers {0, 1, c, (n - 1)}. This suggests the question: Can we perform arithmetic operations within the confines of this set? It turns out that we can; this technique is known as modular arithmetic.

Modular arithmetic exhibits the following properties:

1. [(a mod n) + (b mod n)] mod n = (a + b) mod n 2. [(a mod n) - (b mod n)] mod n = (a - b) mod n 3. [(a mod n) * (b mod n)] mod n = (a * b) mod n

We demonstrate the first property. Define (a mod n) = ra and (b mod n) = rb. Then we can write a = ra + jn for some integer j and b = rb + kn for some integer k. Then

(a + b) mod n = (ra + jn + rb + kn) mod n = (ra + rb + (k + j)n) mod n = (ra + rb) mod n = [(a mod n) + (b mod n)] mod n

The remaining properties are proven as easily. Here are examples of the three properties:

11 mod 8 = 3; 15 mod 8 = 7[(11 mod 8) + (15 mod 8)] mod 8 = 10 mod 8 = 2(11 + 15) mod 8 = 26 mod 8 = 2

[(11 mod 8) - (15 mod 8)] mod 8 = -4 mod 8 = 4(11 - 15) mod 8 = -4 mod 8 = 4

[(11 mod 8) * (15 mod 8)] mod 8 = 21 mod 8 = 5(11 * 15) mod 8 = 165 mod 8 = 5


Exponentiation is performed by repeated multiplication, as in ordinary arith-metic. (We have more to say about exponentiation in Chapter 8.)

To find 117 mod 13, we can proceed as follows:

112 = 121 K 4 (mod 13)

114 = (112)2 K 42 K 3 (mod 13)

117 K 11 * 4 * 3 K 132 K 2 (mod 13)

Thus, the rules for ordinary arithmetic involving addition, subtraction, and multiplication carry over into modular arithmetic.

Table 4.2 provides an illustration of modular addition and multiplication modulo 8. Looking at addition, the results are straightforward, and there is a regu-lar pattern to the matrix. Both matrices are symmetric about the main diagonal in conformance to the commutative property of addition and multiplication. As in ordinary addition, there is an additive inverse, or negative, to each integer in modular arithmetic. In this case, the negative of an integer x is the integer y such that (x + y) mod 8 = 0. To find the additive inverse of an integer in the left-hand column, scan across the corresponding row of the matrix to find the value 0; the

* 0 1 2 3 4 5 6 7

0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7

2 0 2 4 6 0 2 4 6

3 0 3 6 1 4 7 2 5

4 0 4 0 4 0 4 0 4

5 0 5 2 7 4 1 6 3

6 0 6 4 2 0 6 4 2

7 0 7 6 5 4 3 2 1

(b) Multiplication modulo 8

Table 4.2 Arithmetic Modulo 8

+ 0 1 2 3 4 5 6 70 0 1 2 3 4 5 6 71 1 2 3 4 5 6 7 02 2 3 4 5 6 7 0 13 3 4 5 6 7 0 1 24 4 5 6 7 0 1 2 35 5 6 7 0 1 2 3 46 6 7 0 1 2 3 4 57 7 0 1 2 3 4 5 6

(a) Addition modulo 8

w -w w - 1

0 0 —

1 7 1

2 6 —

3 5 3

4 4 —

5 3 5

6 2 —

7 1 7

(c) Additive and multiplicative inverse modulo 8


integer at the top of that column is the additive inverse; thus, (2 + 6) mod 8 = 0. Similarly, the entries in the multiplication table are straightforward. In ordinary arithmetic, there is a multiplicative inverse, or reciprocal, to each integer. In mod-ular arithmetic mod 8, the multiplicative inverse of x is the integer y such that (x * y) mod 8 = 1 mod 8. Now, to find the multiplicative inverse of an integer from the multiplication table, scan across the matrix in the row for that integer to find the value 1; the integer at the top of that column is the multiplicative inverse; thus, (3 * 3) mod 8 = 1. Note that not all integers mod 8 have a multiplicative inverse; more about that later.

Properties of Modular Arithmetic

Define the set Zn as the set of nonnegative integers less than n:

Zn = {0, 1, c, (n - 1)}

This is referred to as the set of residues, or residue classes (mod n). To be more pre-cise, each integer in Zn represents a residue class. We can label the residue classes (mod n) as [0], [1], [2], c, [n - 1], where

[r] = {a: a is an integer, a K r (mod n)}

The residue classes (mod 4) are

[0] = {c, -16, -12, -8, -4, 0, 4, 8, 12, 16, c}

[1] = {c, -15, -11, -7, -3, 1, 5, 9, 13, 17, c}

[2] = {c, -14, -10, -6, -2, 2, 6, 10, 14, 18, c}

[3] = {c, -13, -9, -5, -1, 3, 7, 11, 15, 19, c}

Of all the integers in a residue class, the smallest nonnegative integer is the one used to represent the residue class. Finding the smallest nonnegative integer to which k is congruent modulo n is called reducing k modulo n.

If we perform modular arithmetic within Zn, the properties shown in Table 4.3 hold for integers in Zn. We show in the next section that this implies that Zn is a com-mutative ring with a multiplicative identity element.

Table 4.3 Properties of Modular Arithmetic for Integers in Zn

Property Expression

Commutative Laws(w + x) mod n = (x + w) mod n(w * x) mod n = (x * w) mod n

Associative Laws[(w + x) + y] mod n = [w + (x + y)] mod n[(w * x) * y] mod n = [w * (x * y)] mod n

Distributive Law [w * (x + y)] mod n = [(w * x) + (w * y)] mod n

Identities(0 + w) mod n = w mod n(1 * w) mod n = w mod n

Additive Inverse (- w) For each w ∈ Zn, there exists a z such that w + z K 0 mod n


There is one peculiarity of modular arithmetic that sets it apart from ordinary arithmetic. First, observe that (as in ordinary arithmetic) we can write the following:

if (a + b) K (a + c) (mod n) then b K c (mod n) (4.4)

(5 + 23) K (5 + 7) (mod 8); 23 K 7(mod 8)

Equation (4.4) is consistent with the existence of an additive inverse. Adding the additive inverse of a to both sides of Equation (4.4), we have

((-a) + a + b) K ((-a) + a + c) (mod n)b K c (mod n)

However, the following statement is true only with the attached condition:

if (a * b) K (a * c)(mod n) then b K c (mod n) if a is relatively prime to n (4.5)

Recall that two integers are relatively prime if their only common positive integer factor is 1. Similar to the case of Equation (4.4), we can say that Equation (4.5) is consistent with the existence of a multiplicative inverse. Applying the multiplicative inverse of a to both sides of Equation (4.5), we have

((a- 1)ab) K ((a- 1)ac) (mod n)b K c (mod n)

To see this, consider an example in which the condition of Equation (4.5) does not hold. The integers 6 and 8 are not relatively prime, since they have the com-mon factor 2. We have the following:

6 * 3 = 18 K 2 (mod 8)

6 * 7 = 42 K 2 (mod 8)

Yet 3 [ 7 (mod 8).

The reason for this strange result is that for any general modulus n, a multi-plier a that is applied in turn to the integers 0 through (n - 1) will fail to produce a complete set of residues if a and n have any factors in common.

With a = 6 and n = 8,

Z8 0 1 2 3 4 5 6 7Multiply by 6 0 6 12 18 24 30 36 42Residues 0 6 4 2 0 6 4 2

Because we do not have a complete set of residues when multiplying by 6, more than one integer in Z8 maps into the same residue. Specifically, 6 * 0 mod 8 =6 * 4 mod 8; 6 * 1 mod 8 = 6 * 5 mod 8; and so on. Because this is a many-to-one mapping, there is not a unique inverse to the multiply operation.

(Continued)


In general, an integer has a multiplicative inverse in Zn if that integer is rela-tively prime to n. Table 4.2c shows that the integers 1, 3, 5, and 7 have a multiplicative inverse in Z8; but 2, 4, and 6 do not.

Euclidean Algorithm Revisited

The Euclidean algorithm can be based on the following theorem: For any integers a, b, with a Ú b Ú 0,

gcd(a, b) = gcd(b, a mod b) (4.6)

However, if we take a = 5 and n = 8, whose only common factor is 1,

Z8 0 1 2 3 4 5 6 7Multiply by 5 0 5 10 15 20 25 30 35Residues 0 5 2 7 4 1 6 3

The line of residues contains all the integers in Z8, in a different order.

gcd(55, 22) = gcd(22, 55 mod 22) = gcd(22, 11) = 11

To see that Equation (4.6) works, let d = gcd(a, b). Then, by the definition of gcd, d � a and d � b. For any positive integer b, we can express a as

a = kb + r K r (mod b)a mod b = r

with k, r integers. Therefore, (a mod b) = a - kb for some integer k. But because d � b, it also divides kb. We also have d � a. Therefore, d � (a mod b). This shows that d is a common divisor of b and (a mod b). Conversely, if d is a common divisor of b and (a mod b), then d � kb and thus d � [kb + (a mod b)], which is equivalent to d � a. Thus, the set of common divisors of a and b is equal to the set of common divisors of b and (a mod b). Therefore, the gcd of one pair is the same as the gcd of the other pair, proving the theorem.

Equation (4.6) can be used repetitively to determine the greatest common divisor.

gcd(18, 12) = gcd(12, 6) = gcd(6, 0) = 6

gcd(11, 10) = gcd(10, 1) = gcd(1, 0) = 1

This is the same scheme shown in Equation (4.3), which can be rewritten in the following way.

(Continued)


Euclidean Algorithm

Calculate Which satisfies

r1 = a mod b a = q1b + r1

r2 = b mod r1 b = q2r1 + r2

r3 = r1 mod r2 r1 = q3r2 + r3

• •• •• •

rn = rn - 2 mod rn - 1 rn - 2 = qnrn - 1 + rn

rn + 1 = rn - 1 mod rn = 0 rn - 1 = qn + 1rn + 0 d = gcd(a, b) = rn

We can define the Euclidean algorithm concisely as the following recursive function.Euclid(a,b) if (b=0) then return a; else return Euclid(b, a mod b);

The Extended Euclidean Algorithm

We now proceed to look at an extension to the Euclidean algorithm that will be important for later computations in the area of finite fields and in encryption algo-rithms, such as RSA. For given integers a and b, the extended Euclidean algorithm not only calculate the greatest common divisor d but also two additional integers x and y that satisfy the following equation.

ax + by = d = gcd(a, b) (4.7)

It should be clear that x and y will have opposite signs. Before examining the algorithm, let us look at some of the values of x and y when a = 42 and b = 30. Note that gcd(42, 30) = 6. Here is a partial table of values3 for 42x + 30y.

x y

−3 −2 −1 0 1 2 3

-3 -216 -174 -132 -90 -48 -6 36-2 -186 -144 -102 -60 -18 24 66-1 -156 -114 -72 -30 12 54 96

0 -126 -84 -42 0 42 84 1261 -96 -54 -12 30 72 114 1562 -66 -24 18 60 102 144 1863 -36 6 48 90 132 174 216

Observe that all of the entries are divisible by 6. This is not surpris-ing, because both 42 and 30 are divisible by 6, so every number of the form 42x + 30y = 6(7x + 5y) is a multiple of 6. Note also that gcd(42, 30) = 6 appears in the table. In general, it can be shown that for given integers a and b, the smallest positive value of ax + by is equal to gcd(a, b).

3This example is taken from [SILV06].


Now let us show how to extend the Euclidean algorithm to determine (x, y, d) given a and b. We again go through the sequence of divisions indicated in Equation (4.3), and we assume that at each step i we can find integers xi and yi that satisfy ri = axi + byi. We end up with the following sequence.

a = q1b + r1 r1 = ax1 + by1

b = q2r1 + r2 r2 = ax2 + by2

r1 = q3r2 + r3 r3 = ax3 + by3

~ ~~ ~~ ~

rn - 2 = qnrn - 1 + rn rn = axn + byn

rn - 1 = qn + 1rn + 0

Now, observe that we can rearrange terms to write ri = ri- 2 - ri- 1 qi (4.8)

Also, in rows i - 1 and i - 2, we find the values

ri- 2 = axi- 2 + byi- 2 and ri- 1 = axi- 1 + byi- 1

Substituting into Equation (4.8), we have

ri = (axi- 2 + byi- 2) - (axi- 1 + byi- 1)qi

= a(xi- 2 - qixi- 1) + b(yi- 2 - qiyi- 1)

But we have already assumed that ri = axi + byi. Therefore,

xi = xi- 2 - qixi- 1 and yi = yi- 2 - qiyi- 1

We now summarize the calculations:

Extended Euclidean Algorithm

Calculate Which satisfies Calculate Which satisfies

r- 1 = a x- 1 = 1; y- 1 = 0 a = ax- 1 + by- 1

r0 = b x0 = 0; y0 = 1 b = ax0 + by0

r1 = a mod b q1 = :a/b;

a = q1b + r1 x1 = x- 1 - q1x0 = 1y1 = y- 1 - q1y0 = -q1

r1 = ax1 + by1

r2 = b mod r1 q2 = :b/r1;

b = q2r1 + r2 x2 = x0 - q2x1y2 = y0 - q2y1

r2 = ax2 + by2

r3 = r1 mod r2 r1 = q3r2 + r3 x3 = x1 - q3x2 r3 = ax3 + by3

q3 = :r1/r2; y3 = y1 - q3y2

• • • •• • • •• • • •

rn = rn - 2 mod rn - 1 qn = :rn - 2/rn - 1;

rn - 2 = qnrn - 1 + rn xn = xn - 2 - qnxn - 1 yn = yn - 2 - qnyn - 1

rn = axn + byn

rn + 1 = rn - 1 mod rn = 0qn + 1 = :rn - 1/rn;

rn - 1 = qn + 1rn + 0 d = gcd(a, b) = rn

x = xn; y = yn

4.4 / groups, rings, and Fields 99

We need to make several additional comments here. In each row, we calculate a new remainder ri based on the remainders of the previous two rows, namely ri- 1 and ri - 2. To start the algorithm, we need values for r0 and r- 1, which are just a and b. It is then straightforward to determine the required values for x- 1, y- 1, x0, and y0.

We know from the original Euclidean algorithm that the process ends with a remainder of zero and that the greatest common divisor of a and b is d = gcd(a, b) = rn. But we also have determined that d = rn = axn + byn. Therefore, in Equation (4.7), x = xn and y = yn.

As an example, let us use a = 1759 and b = 550 and solve for 1759x + 550y = gcd(1759, 550). The results are shown in Table 4.4. Thus, we have 1759 * (-111) + 550 * 355= -195249 + 195250 = 1.

4.4 grOups, rings, anD FieLDs

Groups, rings, and fields are the fundamental elements of a branch of mathematics known as abstract algebra, or modern algebra. In abstract algebra, we are concerned with sets on whose elements we can operate algebraically; that is, we can combine two elements of the set, perhaps in several ways, to obtain a third element of the set. These operations are subject to specific rules, which define the nature of the set. By convention, the notation for the two principal classes of operations on set elements is usually the same as the notation for addition and multiplication on ordinary num-bers. However, it is important to note that, in abstract algebra, we are not limited to ordinary arithmetical operations. All this should become clear as we proceed.

Groups

A group G, sometimes denoted by {G, ~}, is a set of elements with a binary opera-tion denoted by ~ that associates to each ordered pair (a, b) of elements in G an ele-ment (a ~ b) in G, such that the following axioms are obeyed:4

Table 4.4 Extended Euclidean Algorithm Example

i ri qi xi yi-1 1759 1 0

0 550 0 1

1 109 3 1 -3

2 5 5 -5 16

3 4 21 106 -339

4 1 1 -111 355

5 0 4

Result: d = 1; x = -111; y = 355

4Theoperator•isgenericandcanrefertoaddition,multiplication,orsomeothermathematicaloperation.

(A1) Closure: If a and b belong to G, then a ~ b is also in G.

(A2) Associative: a ~ (b ~ c) = (a ~ b) ~ c for all a, b, c in G.


If a group has a finite number of elements, it is referred to as a finite group, and the order of the group is equal to the number of elements in the group. Otherwise, the group is an infinite group.

A group is said to be abelian if it satisfies the following additional condition:

(A5) Commutative: a ~ b = b ~ a for all a, b in G.

(A3) Identity element: There is an element e in G such that a ~ e = e ~ a = a for all a in G.

(A4) Inverse element: For each a in G, there is an element a′ in G such that a ~ a′ = a′ ~ a = e.

5This is equivalent to the definition of permutation in Chapter 2, which stated that a permutation of a finite set of elements S is an ordered sequence of all the elements of S, with each element appearing exactly once.

Let Nn denote a set of n distinct symbols that, for convenience, we represent as {1, 2, c, n}. A permutation of n distinct symbols is a one-to-one mapping from Nn to Nn.5 Define Sn to be the set of all permutations of n distinct symbols. Each element of Sn is represented by a permutation of the integers p in 1, 2, c, n. It is easy to demonstrate that Sn is a group:

A1: If (p, r H Sn), then the composite mapping p ~ r is formed by per-muting the elements of r according to the permutation π. For ex-ample, {3, 2, 1} ~ {1, 3, 2} = {2, 3, 1}. Clearly, p ~ r H Sn.

A2: The composition of mappings is also easily seen to be associative.

A3: The identity mapping is the permutation that does not alter the order of the n elements. For Sn, the identity element is {1, 2, c, n}.

A4: For any p H Sn, the mapping that undoes the permutation defined by p is the inverse element for p. There will always be such an inverse. For example {2, 3, 1} ~ {3, 1, 2} = {1, 2, 3}.

The set of integers (positive, negative, and 0) under addition is an abelian group. The set of nonzero real numbers under multiplication is an abelian group. The set Sn from the preceding example is a group but not an abelian group for n 7 2.

When the group operation is addition, the identity element is 0; the inverse ele-ment of a is -a; and subtraction is defined with the following rule: a - b = a + (-b).

CyCliC Group We define exponentiation within a group as a repeated application of the group operator, so that a3 = a ~ a ~ a. Furthermore, we define a0 = e as the identity element, and a- n = (a′)n, where a′ is the inverse element of a within the group. A group G is cyclic if every element of G is a power ak (k is an integer) of

4.4 / groups, rings, and Fields 101

Rings

A ring R, sometimes denoted by {R, + , *}, is a set of elements with two binary operations, called addition and multiplication,6 such that for all a, b, c in R the fol-lowing axioms are obeyed.

(A1–A5) R is an abelian group with respect to addition; that is, R satisfies axioms A1 through A5. For the case of an additive group, we denote the identity element as 0 and the inverse of a as -a.

(M1) Closure under multiplication: If a and b belong to R, then ab is also in R.

(M2) Associativity of multiplication: a(bc) = (ab)c for all a, b, c in R.

(M3) Distributive laws: a(b + c) = ab + ac for all a, b, c in R. (a + b)c = ac + bc for all a, b, c in R.

In essence, a ring is a set in which we can do addition, subtraction [a - b = a + (-b)], and multiplication without leaving the set.

a fixed element a H G. The element a is said to generate the group G or to be a generator of G. A cyclic group is always abelian and may be finite or infinite.

The additive group of integers is an infinite cyclic group generated by the element 1. In this case, powers are interpreted additively, so that n is the nth power of 1.

6Generally, we do not use the multiplication symbol, *, but denote multiplication by the concatenation of two elements.

With respect to addition and multiplication, the set of all n-square matrices over the real numbers is a ring.

Let S be the set of even integers (positive, negative, and 0) under the usual opera-tions of addition and multiplication. S is a commutative ring. The set of all n-square matrices defined in the preceding example is not a commutative ring.

The set Zn of integers {0, 1, c, n - 1}, together with the arithmetic opera-tions modulo n, is a commutative ring (Table 4.3).

A ring is said to be commutative if it satisfies the following additional condition:

(M4) Commutativity of multiplication: ab = ba for all a, b in R.

Next, we define an integral domain, which is a commutative ring that obeys the following axioms.

(M5) Multiplicative identity: There is an element 1 in R such that a1 = 1a = a for all a in R.

(M6) No zero divisors: If a, b in R and ab = 0, then either a = 0 or b = 0.


Fields

A field F, sometimes denoted by {F, +, *}, is a set of elements with two binary op-erations, called addition and multiplication, such that for all a, b, c in F the following axioms are obeyed.

(A1–M6) F is an integral domain; that is, F satisfies axioms A1 through A5 and M1 through M6.

(M7) Multiplicative inverse: For each a in F, except 0, there is an element a- 1 in F such that aa- 1 = (a- 1)a = 1.

In essence, a field is a set in which we can do addition, subtraction, multiplica-tion, and division without leaving the set. Division is defined with the following rule: a/b = a(b- 1).

Let S be the set of integers, positive, negative, and 0, under the usual operations of addition and multiplication. S is an integral domain.

Familiar examples of fields are the rational numbers, the real numbers, and the complex numbers. Note that the set of all integers is not a field, because not every element of the set has a multiplicative inverse; in fact, only the elements 1 and -1 have multiplicative inverses in the integers.

Figure 4.2 summarizes the axioms that define groups, rings, and fields.

4.5 Finite FieLDs OF the FOrm gF(p)

In Section 4.4, we defined a field as a set that obeys all of the axioms of Figure 4.2 and gave some examples of infinite fields. Infinite fields are not of particular inter-est in the context of cryptography. However, finite fields play a crucial role in many cryptographic algorithms. It can be shown that the order of a finite field (number of elements in the field) must be a power of a prime pn, where n is a positive integer. We discuss prime numbers in detail in Chapter 8. Here, we need only say that a prime number is an integer whose only positive integer factors are itself and 1. That is, the only positive integers that are divisors of p are p and 1.

The finite field of order pn is generally written GF(pn); GF stands for Galois field, in honor of the mathematician who first studied finite fields. Two special cases are of interest for our purposes. For n = 1, we have the finite field GF(p); this finite field has a different structure than that for finite fields with n 7 1 and is studied in this section. In Section 4.7, we look at finite fields of the form GF(2n).

Finite Fields of Order p

For a given prime, p, we define the finite field of order p, GF(p), as the set Zp of integers {0, 1, c, p - 1} together with the arithmetic operations modulo p.

4.5 / Finite Fields oF the Form gF(p) 103

Recall that we showed in Section 4.3 that the set Zn of integers {0, 1, c, n - 1}, together with the arithmetic operations modulo n, is a commuta-tive ring (Table 4.3). We further observed that any integer in Zn has a multiplica-tive inverse if and only if that integer is relatively prime to n [see discussion of Equation (4.5)].7 If n is prime, then all of the nonzero integers in Zn are relatively prime to n, and therefore there exists a multiplicative inverse for all of the nonzero integers in Zn. Thus, for Zp we can add the following properties to those listed in Table 4.3:

Multiplicative inverse (w - 1)

For each w H Zp, w ≠ 0, there exists a z H Zp such that w * z K 1 (mod p)

Because w is relatively prime to p, if we multiply all the elements of Zp by w, the resulting residues are all of the elements of Zp permuted. Thus, exactly one of the residues has the value 1. Therefore, there is some integer in Zp that, when multiplied by w, yields the residue 1. That integer is the multiplicative inverse of w, designated w - 1. Therefore, Zp is in fact a finite field. Furthermore, Equation (4.5)

FIELD(A1) Closure under addition: If a and b belong to S, then a + b is also in S(A2) Associativity of addition: a + (b + c) = (a + b) + c for all a, b, c in S(A3) Additive identity: There is an element 0 in R such that a + 0 = 0 + a = a for all a in S(A4) Additive inverse: For each a in S there is an element –a in S such that a + (–a) = (–a) + a = 0

(A5) Commutativity of addition: a + b = b + a for all a, b in S

(M1) Closure under multiplication: If a and b belong to S, then ab is also in S(M2) Associativity of multiplication: a(bc) = (ab)c for all a, b, c in S(M3) Distributive laws: a(b + c) = ab + ac for all a, b, c in S (a + b)c = ac + bc for all a, b, c in S

(M4) Commutativity of multiplication: ab = ba for all a, b in S

(M5) Multiplicative identity: There is an element 1 in S such that a1 = 1a = a for all a in S(M6) No zero divisors: If a, b in S and ab = 0, then either a = 0 or b = 0

(M7) Multiplicative inverse: If a belongs to S and a 0, there is an element a–1 in S such that aa–1 = a–1a = 1

Integral Domain

Commutative Ring

Ring

Abelian Group

Group

Figure 4.2 Group, Ring, and Field

7As stated in the discussion of Equation (4.5), two integers are relatively prime if their only common positive integer factor is 1.


is consistent with the existence of a multiplicative inverse and can be rewritten with-out the condition:

if (a * b) K (a * c)(mod p) then b K c (mod p) (4.9)

Multiplying both sides of Equation (4.9) by the multiplicative inverse of a, we have

((a- 1) * a * b) K ((a- 1) * a * c) (mod p)

b K c (mod p)

The simplest finite field is GF(2). Its arithmetic operations are easily summarized:

+ 0 10 0 11 1 0

* 0 10 0 01 0 1

w -w w-1

0 0 9

1 1 1

Addition Multiplication Inverses

In this case, addition is equivalent to the exclusive-OR (XOR) operation, and multiplication is equivalent to the logical AND operation.

Table 4.5 shows arithmetic operations in GF(7). This is a field of order 7 using modular arithmetic modulo 7. As can be seen, it satisfies all of the properties re-quired of a field (Figure 4.2). Compare this table with Table 4.2. In the latter case, we see that the set Z8, using modular arithmetic modulo 8, is not a field. Later in this chapter, we show how to define addition and multiplication operations on Z8 in such a way as to form a finite field.

Finding the Multiplicative Inverse in GF(p)

It is easy to find the multiplicative inverse of an element in GF(p) for small values of p. You simply construct a multiplication table, such as shown in Table 4.5b, and the desired result can be read directly. However, for large values of p, this approach is not practical.

If a and b are relatively prime, then b has a multiplicative inverse modulo a. That is, if gcd(a, b) = 1, then b has a multiplicative inverse modulo a. That is, for positive integer b 6 a, there exists a b- 1 6 a such that bb- 1 = 1 mod a. If a is a prime number and b 6 a, then clearly a and b are relatively prime and have a great-est common divisor of 1. We now show that we can easily compute b- 1 using the extended Euclidean algorithm.

We repeat here Equation (4.7), which we showed can be solved with the ex-tended Euclidean algorithm:

ax + by = d = gcd(a, b)

Now, if gcd(a, b) = 1, then we have ax + by = 1. Using the basic equalities of modular arithmetic, defined in Section 4.3, we can say

[(ax mod a) + (by mod a)] mod a = 1 mod a

0 + (by mod a) = 1

But if by mod a = 1, then y = b- 1. Thus, applying the extended Euclidean algorithm to Equation (4.7) yields the value of the multiplicative inverse of b if gcd(a, b) = 1. Consider the example that was shown in Table 4.4. Here we have a = 1759, which is a prime number, and b = 550. The solution of the equation 1759x + 550y = d yields a value of y = 355. Thus, b- 1 = 355. To verify, we calcu-late 550 * 355 mod 1759 = 195250 mod 1759 = 1.

More generally, the extended Euclidean algorithm can be used to find a multiplicative inverse in Zn for any n. If we apply the extended Euclidean algo-rithm to the equation nx + by = d, and the algorithm yields d = 1, then y = b- 1 in Zn.

Summary

In this section, we have shown how to construct a finite field of order p, where p is prime. Specifically, we defined GF(p) with the following properties.

1. GF(p) consists of p elements.

2. The binary operations + and * are defined over the set. The operations of addition, subtraction, multiplication, and division can be performed without leaving the set. Each element of the set other than 0 has a multiplicative inverse.

We have shown that the elements of GF(p) are the integers {0, 1, c, p - 1} and that the arithmetic operations are addition and multiplication mod p.

Table 4.5 Arithmetic in GF(7)

+ 0 1 2 3 4 5 6

0 0 1 2 3 4 5 6

1 1 2 3 4 5 6 0

2 2 3 4 5 6 0 1

3 3 4 5 6 0 1 2

4 4 5 6 0 1 2 3

5 5 6 0 1 2 3 4

6 6 0 1 2 3 4 5

(a) Addition modulo 7

* 0 1 2 3 4 5 6

0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6

2 0 2 4 6 1 3 5

3 0 3 6 2 5 1 4

4 0 4 1 5 2 6 3

5 0 5 3 1 6 4 2

6 0 6 5 4 3 2 1

(b) Multiplication modulo 7

w -w w - 1

0 0 –

1 6 1

2 5 4

3 4 5

4 3 2

5 2 3

6 1 6

(c) Additive and multiplicative inverses modulo 7

4.5 / Finite Fields oF the Form gF(p) 105


4.6 pOLynOmiaL arithmetic

Before continuing our discussion of finite fields, we need to introduce the interest-ing subject of polynomial arithmetic. We are concerned with polynomials in a single variable x, and we can distinguish three classes of polynomial arithmetic.

• Ordinary polynomial arithmetic, using the basic rules of algebra.

• Polynomial arithmetic in which the arithmetic on the coefficients is performed modulo p; that is, the coefficients are in GF(p).

• Polynomial arithmetic in which the coefficients are in GF(p), and the polynomials are defined modulo a polynomial m(x) whose highest power is some integer n.

This section examines the first two classes, and the next section covers the last class.

Ordinary Polynomial Arithmetic

A polynomial of degree n (integer n Ú 0) is an expression of the form

f(x) = anxn + an - 1xn - 1 + P + a1x + a0 = a

n

i = 0aix

i

where the ai are elements of some designated set of numbers S, called the coefficient set, and an ≠ 0. We say that such polynomials are defined over the coefficient set S.

A zero-degree polynomial is called a constant polynomial and is simply an element of the set of coefficients. An nth-degree polynomial is said to be a monic polynomial if an = 1.

In the context of abstract algebra, we are usually not interested in evaluating a polynomial for a particular value of x [e.g., f(7)]. To emphasize this point, the vari-able x is sometimes referred to as the indeterminate.

Polynomial arithmetic includes the operations of addition, subtraction, and multiplication. These operations are defined in a natural way as though the variable x was an element of S. Division is similarly defined, but requires that S be a field. Examples of fields include the real numbers, rational numbers, and Zp for p prime. Note that the set of all integers is not a field and does not support polynomial division.

Addition and subtraction are performed by adding or subtracting correspond-ing coefficients. Thus, if

f(x) = an

i = 0aix

i; g(x) = am

i = 0bix

i; n Ú m

then addition is defined as

f(x) + g(x) = am

i = 0(ai + bi)xi + a

n

i = m + 1aix

i

and multiplication is defined as

f(x) * g(x) = an + m

i = 0cix

i

4.6 / polynomial arithmetiC 107

where

ck = a0bk + a1bk - 1 + P + ak - 1b1 + akb0

In the last formula, we treat ai as zero for i 7 n and bi as zero for i 7 m. Note that the degree of the product is equal to the sum of the degrees of the two polynomials.

x3

x3

+ +x2

+2x2

x2 x

2

+–+ ( )

× ( )

– ( )

x–

1

+ 3

(a) Addition

(d) Division(c) Multiplication

x3

x3

+ +x2

+ x2

x2 x

2

x3

x 2

+

+

+x2

x3 – x2

2x2

+ x

– x

x

2

+ 2

2x2– 2x + 2

x4 –– –x3 2x

– 2x

x5 + +x4 2x2

x5 +3x2

+– 1 x2 x +– 1

+ 2

+ 2

x3

x3

+ +x2

x2 x

2

+–

x+

1

+ 1

(b) Subtraction

Figure 4.3 Examples of Polynomial Arithmetic

8In fact, the set of polynomials whose coefficients are elements of a commutative ring forms a polynomial ring, but that is of no interest in the present context.

As an example, let f(x) = x3 + x2 + 2 and g(x) = x2 - x + 1, where S is the set of integers. Then

f(x) + g(x) = x3 + 2x2 - x + 3

f(x) - g(x) = x3 + x + 1

f(x) * g(x) = x5 + 3x2 - 2x + 2

Figures 4.3a through 4.3c show the manual calculations. We comment on division subsequently.

Polynomial Arithmetic with Coefficients in Zp

Let us now consider polynomials in which the coefficients are elements of some field F; we refer to this as a polynomial over the field F. In that case, it is easy to show that the set of such polynomials is a ring, referred to as a polynomial ring. That is, if we consider each distinct polynomial to be an element of the set, then that set is a ring.8


Now, if we attempt to perform polynomial division over a coefficient set that is not a field, we find that division is not always defined.

When polynomial arithmetic is performed on polynomials over a field, then division is possible. Note that this does not mean that exact division is possible. Let us clarify this distinction. Within a field, given two elements a and b, the quotient a/b is also an element of the field. However, given a ring R that is not a field, in general, division will result in both a quotient and a remainder; this is not exact division.

Consider the division 5/3 within a set S. If S is the set of rational numbers, which is a field, then the result is simply expressed as 5/3 and is an element of S. Now suppose that S is the field Z7 . In this case, we calculate (using Table 4.5c)

5/3 = (5 * 3- 1) mod 7 = (5 * 5) mod 7 = 4

which is an exact solution. Finally, suppose that S is the set of integers, which is a ring but not a field. Then 5/3 produces a quotient of 1 and a remainder of 2:

5/3 = 1 + 2/3

5 = 1 * 3 + 2

Thus, division is not exact over the set of integers.

If the coefficient set is the integers, then (5x2)>(3x) does not have a solution, because it would require a coefficient with a value of 5/3, which is not in the coefficient set. Suppose that we perform the same polynomial division over Z7. Then we have (5x2)>(3x) = 4x, which is a valid polynomial over Z7.

However, as we demonstrate presently, even if the coefficient set is a field, polynomial division is not necessarily exact. In general, division will produce a quo-tient and a remainder. We can restate the division algorithm of Equation (4.1) for polynomials over a field as follows. Given polynomials f(x) of degree n and g(x) of degree (m), (n Ú m), if we divide f(x) by g(x), we get a quotient q(x) and a remain-der r(x) that obey the relationship

f(x) = q(x)g(x) + r(x) (4.10)

with polynomial degrees:

Degree f(x) = nDegree g(x) = mDegree q(x) = n - mDegree r(x) … m - 1

With the understanding that remainders are allowed, we can say that polyno-mial division is possible if the coefficient set is a field.


For our purposes, polynomials over GF(2) are of most interest. Recall from Section 4.5 that in GF(2), addition is equivalent to the XOR operation, and multiplica-tion is equivalent to the logical AND operation. Further, addition and subtraction are equivalent mod 2: 1 + 1 = 1 - 1 = 0; 1 + 0 = 1 - 0 = 1; 0 + 1 = 0 - 1 = 1.

In an analogy to integer arithmetic, we can write f(x) mod g(x) for the remain-der r(x) in Equation (4.10). That is, r(x) = f(x) mod g(x). If there is no remainder [i.e., r(x) = 0], then we can say g(x) divides f(x), written as g(x) � f(x). Equivalently, we can say that g(x) is a factor of f(x) or g(x) is a divisor of f(x).

For the preceding example [f(x) = x3 + x2 + 2 and g(x) = x2 - x + 1], f(x)>g(x) produces a quotient of q(x) = x + 2 and a remainder r(x) = x, as shown in Figure 4.3d. This is easily verified by noting that

q(x)g(x) + r(x) = (x + 2)(x2 - x + 1) + x = (x3 + x2 - x + 2) + x

= x3 + x2 + 2 = f(x)

Figure 4.4 shows an example of polynomial arithmetic over GF(2). For f(x) = (x7 + x5 + x4 + x3 + x + 1) and g(x) = (x3 + x + 1), the figure shows f(x) + g(x); f(x) - g(x); f(x) * g(x); and f(x)/g(x). Note that g(x) � f(x).

A polynomial f(x) over a field F is called irreducible if and only if f(x) can-not be expressed as a product of two polynomials, both over F, and both of degree lower than that of f(x). By analogy to integers, an irreducible polynomial is also called a prime polynomial.

The polynomial9 f(x) = x4 + 1 over GF(2) is reducible, becausex4 + 1 = (x + 1)(x3 + x2 + x + 1).

Consider the polynomial f(x) = x3 + x + 1. It is clear by inspection that x is not a factor of f(x). We easily show that x + 1 is not a factor of f(x):

x2 + xx + 1� x3 + x + 1

x3 + x2

x2 + xx2 + x

1Thus, f(x) has no factors of degree 1. But it is clear by inspection that if f(x) is re-ducible, it must have one factor of degree 2 and one factor of degree 1. Therefore, f(x) is irreducible.

9In the remainder of this chapter, unless otherwise noted, all examples are of polynomials over GF(2).


Finding the Greatest Common Divisor

We can extend the analogy between polynomial arithmetic over a field and integer arithmetic by defining the greatest common divisor as follows. The polynomial c(x) is said to be the greatest common divisor of a(x) and b(x) if the following are true.

1. c(x) divides both a(x) and b(x).

2. Any divisor of a(x) and b(x) is a divisor of c(x).

An equivalent definition is the following: gcd[a(x), b(x)] is the polynomial of maximum degree that divides both a(x) and b(x).

We can adapt the Euclidean algorithm to compute the greatest common divisor of two polynomials. The equality in Equation (4.6) can be rewritten as the following theorem.

(a) Addition

(c) Multiplication

(d) Division

x4x5 ++x7

xx3

x3x4 ++x5 ++x7 +x 1

+++ ( )1

x3x4 ++x5 ++x7 +x 1

x4x5 ++x7

x3 x

x3 ++ +x 1

+ 1

x5x6 ++x8 x4 ++ +x2

+ x2

x

x7x8 ++x10 x6 ++ +x4

x10 + x4

x3

++× ( )1

x3x4 ++x5 ++x7

x4x5 ++x7

+x

x3 x

1

++– ( )1

(b) Subtraction

x3x4 ++x5 ++

++

x7

x4x5x7

+x 1

x3 + +x 1

x3 + +x 1

x4 1+

x3 x ++ 1

Figure 4.4 Examples of Polynomial Arithmetic over GF(2)


gcd[a(x), b(x)] = gcd[b(x), a(x) mod b(x)] (4.11)

Equation (4.11) can be used repetitively to determine the greatest common divisor. Compare the following scheme to the definition of the Euclidean algorithm for integers.

Euclidean Algorithm for Polynomials

Calculate Which satisfies

r1(x) = a(x) mod b(x) a(x) = q1(x)b(x) + r1(x)

r2(x) = b(x) mod r1(x) b(x) = q2(x)r1(x) + r2(x)

r3(x) = r1(x) mod r2(x) r1(x) = q3(x)r2(x) + r3(x)

• •

• •

• •

rn(x) = rn - 2(x) mod rn - 1(x) rn - 2(x) = qn(x)rn - 1(x) + rn(x)

rn + 1(x) = rn - 1(x) mod rn(x) = 0rn - 1(x) = qn + 1(x)rn(x) + 0

d(x) = gcd(a(x), b(x)) = rn(x)

At each iteration, we have d(x) = gcd(ri+ 1(x), ri(x)) until finally d(x) = gcd(rn(x), 0) = rn(x). Thus, we can find the greatest common divisor of two integers by repetitive application of the division algorithm. This is the Euclidean algorithm for polynomials. The algorithm assumes that the degree of a(x) is greater than the degree of b(x).

Find gcd[a(x), b(x)] for a(x) = x6 + x5 + x4 + x3 + x2 + x + 1 and b(x) =x4 + x2 + x + 1. First, we divide a(x) by b(x):

x2 + x

x4 + x2 + x + 1��x6 + x5 + x4 + x3 + x2 + x + 1x6 + x4 + x3 + x2

x5 + x + 1x5 x3 + x2 + x

x3 + x2 + 1

This yields r1(x) = x3 + x2 + 1 and q1 (x) = x2 + x.

Then, we divide b(x) by r1(x).

x + 1x3 + x2 + 1��x4 + x2 + x + 1

x4 + x3 + xx3 + x2 + 1x3 + x2 + 1

This yields r2(x) = 0 and q2(x) = x + 1.

Therefore, gcd[a(x), b(x)] = r1(x) = x3 + x2 + 1.


Summary

We began this section with a discussion of arithmetic with ordinary polynomials. In ordinary polynomial arithmetic, the variable is not evaluated; that is, we do not plug a value in for the variable of the polynomials. Instead, arithmetic operations are performed on polynomials (addition, subtraction, multiplication, division) using the ordinary rules of algebra. Polynomial division is not allowed unless the coeffi-cients are elements of a field.

Next, we discussed polynomial arithmetic in which the coefficients are ele-ments of GF(p). In this case, polynomial addition, subtraction, multiplication, and division are allowed. However, division is not exact; that is, in general division re-sults in a quotient and a remainder.

Finally, we showed that the Euclidean algorithm can be extended to find the greatest common divisor of two polynomials whose coefficients are elements of a field.

All of the material in this section provides a foundation for the following sec-tion, in which polynomials are used to define finite fields of order pn.

4.7 Finite FieLDs OF the FOrm gF(2n)

Earlier in this chapter, we mentioned that the order of a finite field must be of the form pn, where p is a prime and n is a positive integer. In Section 4.5, we looked at the spe-cial case of finite fields with order p. We found that, using modular arithmetic in Zp, all of the axioms for a field (Figure 4.2) are satisfied. For polynomials over pn, with n 7 1, operations modulo pn do not produce a field. In this section, we show what structure satisfies the axioms for a field in a set with pn elements and concentrate on GF(2n).

Motivation

Virtually all encryption algorithms, both symmetric and public key, involve arithme-tic operations on integers. If one of the operations that is used in the algorithm is division, then we need to work in arithmetic defined over a field. For convenience and for implementation efficiency, we would also like to work with integers that fit exactly into a given number of bits with no wasted bit patterns. That is, we wish to work with integers in the range 0 through 2n - 1, which fit into an n-bit word.

Suppose we wish to define a conventional encryption algorithm that operates on data 8 bits at a time, and we wish to perform division. With 8 bits, we can repre-sent integers in the range 0 through 255. However, 256 is not a prime number, so that if arithmetic is performed in Z256 (arithmetic modulo 256), this set of inte-gers will not be a field. The closest prime number less than 256 is 251. Thus, the set Z251, using arithmetic modulo 251, is a field. However, in this case the 8-bit patterns representing the integers 251 through 255 would not be used, resulting in inefficient use of storage.

4.7 / Finite Fields oF the Form gF(2n) 113

As the preceding example points out, if all arithmetic operations are to be used and we wish to represent a full range of integers in n bits, then arithmetic modulo 2n will not work. Equivalently, the set of integers modulo 2n for n 7 1, is not a field. Furthermore, even if the encryption algorithm uses only addition and multiplication, but not division, the use of the set Z2n is questionable, as the follow-ing example illustrates.

Suppose we wish to use 3-bit blocks in our encryption algorithm and use only the operations of addition and multiplication. Then arithmetic modulo 8 is well defined, as shown in Table 4.2. However, note that in the multiplication table, the nonzero integers do not appear an equal number of times. For example, there are only four occurrences of 3, but twelve occurrences of 4. On the other hand, as was mentioned, there are finite fields of the form GF(2n), so there is in particular a finite field of order 23 = 8. Arithmetic for this field is shown in Table 4.6. In this case, the number of occurrences of the nonzero integers is uniform for multiplication. To summarize,

Integer 1 2 3 4 5 6 7Occurrences in Z8 4 8 4 12 4 8 4Occurrences in GF(23) 7 7 7 7 7 7 7

For the moment, let us set aside the question of how the matrices of Table 4.6 were constructed and instead make some observations.

1. The addition and multiplication tables are symmetric about the main diagonal, in conformance to the commutative property of addition and multiplication. This property is also exhibited in Table 4.2, which uses mod 8 arithmetic.

2. All the nonzero elements defined by Table 4.6 have a multiplicative inverse, unlike the case with Table 4.2.

3. The scheme defined by Table 4.6 satisfies all the requirements for a finite field. Thus, we can refer to this scheme as GF(23).

4. For convenience, we show the 3-bit assignment used for each of the elements of GF(23).

Intuitively, it would seem that an algorithm that maps the integers unevenly onto themselves might be cryptographically weaker than one that provides a uni-form mapping. Thus, the finite fields of the form GF(2n) are attractive for crypto-graphic algorithms.

To summarize, we are looking for a set consisting of 2n elements, together with a definition of addition and multiplication over the set that define a field. We can assign a unique integer in the range 0 through 2n - 1 to each element of the set.


Table 4.6 Arithmetic in GF(23)

000 001 010 011 100 101 110 111

+ 0 1 2 3 4 5 6 7

000 0 0 1 2 3 4 5 6 7001 1 1 0 3 2 5 4 7 6010 2 2 3 0 1 6 7 4 5011 3 3 2 1 0 7 6 5 4100 4 4 5 6 7 0 1 2 3101 5 5 4 7 6 1 0 3 2110 6 6 7 4 5 2 3 0 1111 7 7 6 5 4 3 2 1 0

(a) Addition

000 001 010 011 100 101 110 111

* 0 1 2 3 4 5 6 7

000 0 0 0 0 0 0 0 0 0

001 1 0 1 2 3 4 5 6 7

010 2 0 2 4 6 3 1 7 5

011 3 0 3 6 5 7 4 1 2

100 4 0 4 3 7 6 2 5 1

101 5 0 5 1 4 2 7 3 6

110 6 0 6 7 1 5 3 2 4

111 7 0 7 5 2 1 6 4 3

(b) Multiplication

w -w w - 1

0 0 —

1 1 1

2 2 5

3 3 6

4 4 7

5 5 2

6 6 3

7 7 4

(c) Additive and multiplicative inverses

Keep in mind that we will not use modular arithmetic, as we have seen that this does not result in a field. Instead, we will show how polynomial arithmetic provides a means for constructing the desired field.

Modular Polynomial Arithmetic

Consider the set S of all polynomials of degree n - 1 or less over the field Zp. Thus, each polynomial has the form

f(x) = an - 1xn - 1 + an - 2x

n - 2 + g+ a1x + a0 = an - 1

i = 0aix

i

where each ai takes on a value in the set {0, 1, c, p - 1}. There are a total of pn different polynomials in S.


With the appropriate definition of arithmetic operations, each such set S is a finite field. The definition consists of the following elements.

1. Arithmetic follows the ordinary rules of polynomial arithmetic using the basic rules of algebra, with the following two refinements.

2. Arithmetic on the coefficients is performed modulo p. That is, we use the rules of arithmetic for the finite field Zp.

3. If multiplication results in a polynomial of degree greater than n - 1, then the polynomial is reduced modulo some irreducible polynomial m(x) of degree n. That is, we divide by m(x) and keep the remainder. For a polynomial f(x), the remainder is expressed as r(x) = f(x) mod m(x).

For p = 3 and n = 2, the 32 = 9 polynomials in the set are

0 x 2x1 x + 1 2x + 12 x + 2 2x + 2

For p = 2 and n = 3, the 23 = 8 polynomials in the set are

0 x + 1 x2 + x 1 x2 x2 + x + 1x x2 + 1

The Advanced Encryption Standard (AES) uses arithmetic in the finite field GF(28), with the irreducible polynomial m(x) = x8 + x4 + x3 + x + 1. Consider the two polynomials f(x) = x6 + x4 + x2 + x + 1 and g(x) = x7 + x + 1. Then

f(x) + g(x) = x6 + x4 + x2 + x + 1 + x7 + x + 1 = x7 + x6 + x4 + x2

f(x) * g(x) = x13 + x11 + x9 + x8 + x7

+ x7 + x5 + x3 + x2 + x + x6 + x4 + x2 + x + 1

= x13 + x11 + x9 + x8 + x6 + x5 + x4 + x3 + 1

x5 + x3

x8 + x4 + x3 + x + 1�x13 + x11 + x9 + x8 + x6 + x5 + x4 + x3 + 1x13 + x9 + x8 + x6 + x5

x11 + x4 + x3

x11 + x7 + x6 + x4 + x3

x7 + x6 + 1

Therefore, f(x) * g(x) mod m(x) = x7 + x6 + 1.


It can be shown that the set of all polynomials modulo an irreducible nth-degree polynomial m(x) satisfies the axioms in Figure 4.2, and thus forms a finite field. Furthermore, all finite fields of a given order are isomorphic; that is, any two finite-field structures of a given order have the same structure, but the representation or labels of the elements may be different.

As with ordinary modular arithmetic, we have the notion of a set of residues in modular polynomial arithmetic. The set of residues modulo m(x), an nth-degree polynomial, consists of pn elements. Each of these elements is represented by one of the pn polynomials of degree m 6 n.

The residue class [x + 1], (mod m(x)), consists of all polynomials a(x) such that a(x) K (x + 1) (mod m(x)). Equivalently, the residue class [x + 1] consists of all polynomials a(x) that satisfy the equality a(x) mod m(x) = x + 1.

To construct the finite field GF(23), we need to choose an irreducible poly-nomial of degree 3. There are only two such polynomials: (x3 + x2 + 1) and (x3 + x + 1). Using the latter, Table 4.7 shows the addition and multiplication tables for GF(23). Note that this set of tables has the identical structure to those of Table 4.6. Thus, we have succeeded in finding a way to define a field of order 23.

We can now read additions and multiplications from the table easily. For example, consider binary 100 + 010 = 110. This is equivalent to x2 + x. Also consider 100 * 010 = 011, which is equivalent to x2 * x = x3 and reduces to x + 1. That is, x3 mod (x3 + x + 1) = x + 1, which is equivalent to 011.

Finding the Multiplicative Inverse

Just as the Euclidean algorithm can be adapted to find the greatest common divi-sor of two polynomials, the extended Euclidean algorithm can be adapted to find the multiplicative inverse of a polynomial. Specifically, the algorithm will find the multiplicative inverse of b(x) modulo a(x) if the degree of b(x) is less than the de-gree of a(x) and gcd[a(x), b(x)] = 1. If a(x) is an irreducible polynomial, then it has no factor other than itself or 1, so that gcd[a(x), b(x)] = 1. The algorithm can be characterized in the same way as we did for the extended Euclidean algorithm for integers. Given polynomials a(x) and b(x) with the degree of a(x) greater than the degree of b(x), we wish to solve the following equation for the values v(x), w(x), and d(x), where d(x) = gcd[a(x), b(x)]:

a(x)v(x) + b(x)w(x) = d(x)

If d(x) = 1, then w(x) is the multiplicative inverse of b(x) modulo a(x). The calcula-tions are as follows.

Table 4.7 Polynomial Arithmetic Modulo (x3 + x + 1)

000 001 010 011 100 101 110 111

+ 0 1 x x + 1 x2 x2 + 1 x2 + x x2 + x + 1

000 0 0 1 x x + 1 x2 x2 + 1 x2 + 1 x2 + x + 1

001 1 1 0 x + 1 x x2 + 1 x2 x2 + x + 1 x2 + x

010 x x x + 1 0 1 x2 + x x2 + x + 1 x2 x2 + 1

011 x + 1 x + 1 x 1 0 x2 + x + 1 x2 + x x2 + 1 x2

100 x2 x2 x2 + 1 x2 + x x2 + x + 1 0 1 x x + 1

101 x2 + 1 x2 + 1 x2 x2 + x + 1 x2 + x 1 0 x + 1 x

110 x2 + x x2 + x x2 + x + 1 x2 x2 + 1 x x + 1 0 1

111 x2 + x + 1 x2 + x + 1 x2 + x x2 + 1 x2 x + 1 x 1 0

(a) Addition

000 001 010 011 100 101 110 111

* 0 1 x x + 1 x2 x2 + 1 x2 + x x2 + x + 1000 0 0 0 0 0 0 0 0 0

001 1 0 1 x x + 1 x2 x2 + 1 x2 + x x2 + x + 1010 x 0 x x2 x2 + x x + 1 1 x2 + x + 1 x2 + 1011 x + 1 0 x + 1 x2 + x x2 + 1 x2 + x + 1 x2 1 x

100 x2 0 x2 x + 1 x2 + x + 1 x2 + x x x2 + 1 1

101 x2 + 1 0 x2 + 1 1 x2 x x2 + x + 1 x + 1 x2 + x110 x2 + x 0 x2 + x x2 + x + 1 1 x2 + 1 x + 1 x x2

111 x2 + x + 1 0 x2 + x + 1 x2 + 1 x 1 x2 + 1 x2 x + 1

(b) Multiplication117


Table 4.8 shows the calculation of the multiplicative inverse of (x7 + x + 1) mod (x8 + x4 + x3 + x + 1). The result is that (x7 + x + 1)- 1 = (x7). That is, (x7 + x + 1)(x7) K 1( mod (x8 + x4 + x3 + x + 1)).

Extended Euclidean Algorithm for Polynomials

Calculate Which satisfies Calculate Which satisfies

r- 1(x) = a(x) v- 1(x) = 1; w- 1(x) = 0 a(x) = a(x)v- 1(x) +bw- 1(x)

r0(x) = b(x) v0(x) = 0; w0(x) = 1 b(x) = a(x)v0(x) +b(x)w0(x)

r1(x) = a(x) mod b(x)q1(x) = quotient ofa(x)/b(x)

a(x) = q1(x)b(x) + r1(x)

v1(x) = v- 1(x) - q1(x)v0(x) = 1 w1(x) = w- 1(x) - q1(x)w0(x) = - q1(x)

r1(x) = a(x)v1(x) +b(x)w1(x)

r2(x) = b(x) mod r1(x)q2(x) = quotient ofb(x)/r1(x)

b(x) = q2(x)r1(x) + r2(x)

v2(x) = v0(x) - q2(x)v1(x) w2(x) = w0(x) - q2(x)w1(x)

r2(x) = a(x)v2(x) +b(x)w2(x)

r3(x) = r1(x) mod r2(x)q3(x) = quotient ofr1(x)/r2(x)

r1(x) = q3(x)r2(x) + r3(x)

v3(x) = v1(x) - q3(x)v2(x) w3(x) = w1(x) - q3(x)w2(x)

r3(x) = a(x)v3(x) +b(x)w3(x)

• • • •

• • • •

• • • •

rn(x) = rn - 2(x) mod rn - 1(x) qn(x) = quotient of rn - 2(x)/rn - 3(x)

rn - 2(x) = qn(x)rn - 1(x) + rn(x)

vn(x) = vn - 2(x) - qn(x)vn - 1(x) wn(x) = wn - 2(x) - qn(x)wn - 1(x)

rn(x) = a(x)vn(x) +b(x)wn(x)

rn + 1(x) = rn - 1(x) mod rn(x) = 0 qn + 1(x) = quotient of rn - 1(x)/rn - 2(x)

rn - 1(x) = qn + 1(x)rn(x) + 0

d(x) = gcd(a(x), b(x)) = rn(x) v(x) = vn(x); w(x) =wn(x)

Computational Considerations

A polynomial f(x) in GF(2n)

f(x) = an - 1xn - 1 + an - 2x

n - 2 + g+ a1x + a0 = an - 1

i = 0aix

i

can be uniquely represented by the sequence of its n binary coefficients (an - 1, an - 2, c, a0). Thus, every polynomial in GF(2n) can be represented by an n-bit number.


Addition We have seen that addition of polynomials is performed by adding cor-responding coefficients, and, in the case of polynomials over Z2, addition is just the XOR operation. So, addition of two polynomials in GF(2n) corresponds to a bitwise XOR operation.

Table 4.8 Extended Euclid [(x8 + x4 + x3 + x + 1), (x7 + x + 1)]

Initialization a(x) = x8 + x4 + x3 + x + 1; v- 1(x) = 1; w- 1(x) = 0b(x) = x7 + x + 1; v0(x) = 0; w0(x) = 1

Iteration 1 q1(x) = x; r1(x) = x4 + x3 + x2 + 1 v1(x) = 1; w1(x) = x

Iteration 2 q2(x) = x3 + x2 + 1; r2(x) = x v2(x) = x3 + x2 + 1; w2(x) = x4 + x3 + x + 1

Iteration 3 q3(x) = x3 + x2 + x; r3(x) = 1 v3(x) = x6 + x2 + x + 1; w3(x) = x7

Iteration 4 q4(x) = x; r4(x) = 0 v4(x) = x7 + x + 1; w4(x) = x8 + x4 + x3 + x + 1

Result d(x) = r3(x) = gcd(a(x), b(x)) = 1w(x) = w3(x) = (x7 + x + 1)- 1 mod (x8 + x4 + x3 + x + 1) = x7

Tables 4.6 and 4.7 show the addition and multiplication tables for GF(23) modulo m(x) = (x3 + x + 1). Table 4.6 uses the binary representation, and Table 4.7 uses the polynomial representation.

Consider the two polynomials in GF(28) from our earlier example:

f(x) = x6 + x4 + x2 + x + 1 and g(x) = x7 + x + 1.

(x6 + x4 + x2 + x + 1) + (x7 + x + 1) = x7 + x6 + x4 + x2 (polynomial notation)

(01010111) ⊕ (10000011) = (11010100) (binary notation)

{57} ⊕ {83} = {D4} (hexadecimal notation)10

10A basic refresher on number systems (decimal, binary, hexadecimal) can be found at the Computer Science Student Resource Site at WilliamStallings.com/StudentSupport.html. Here each of two groups of 4 bits in a byte is denoted by a single hexadecimal character, and the two characters are enclosed in brackets.

MultipliCAtion There is no simple XOR operation that will accomplish multi-plication in GF(2n). However, a reasonably straightforward, easily implemented technique is available. We will discuss the technique with reference to GF(28) using m(x) = x8 + x4 + x3 + x + 1, which is the finite field used in AES. The technique readily generalizes to GF(2n).

The technique is based on the observation that

x8 mod m(x) = [m(x) - x8] = (x4 + x3 + x + 1) (4.12)


A moment’s thought should convince you that Equation (4.12) is true; if you are not sure, divide it out. In general, in GF(2n) with an nth-degree polynomial p(x), we have xn mod p(x) = [p(x) - xn].

Now, consider a polynomial in GF(28), which has the form f(x) = b7x7 +

b6x6 + b5x

5 + b4x4 + b3x

3 + b2x2 + b1x + b0. If we multiply by x, we have

x * f(x) = (b7x8 + b6x

7 + b5x6 + b4x

5 + b3x4

+ b2x3 + b1x

2 + b0x) mod m(x) (4.13)

If b7 = 0, then the result is a polynomial of degree less than 8, which is already in reduced form, and no further computation is necessary. If b7 = 1, then reduction modulo m(x) is achieved using Equation (4.12):

x * f(x) = (b6x7 + b5x

6 + b4x5 + b3x

4 + b2x3 + b1x

2 + b0x)

+ (x4 + x3 + x + 1)

It follows that multiplication by x (i.e., 00000010) can be implemented as a 1-bit left shift followed by a conditional bitwise XOR with (00011011), which represents (x4 + x3 + x + 1). To summarize,

x * f(x) = e (b6b5b4b3b2b1b00) if b7 = 0(b6b5b4b3b2b1b00) ⊕ (00011011) if b7 = 1

(4.14)

Multiplication by a higher power of x can be achieved by repeated application of Equation (4.14). By adding intermediate results, multiplication by any constant in GF(28) can be achieved.

In an earlier example, we showed that for f(x) = x6 + x4 + x2 + x + 1, g(x) = x7 + x + 1, and m(x) = x8 + x4 + x3 + x + 1, we have f(x) * g(x) mod m(x) = x7 + x6 + 1. Redoing this in binary arithmetic, we need to compute (01010111) * (10000011). First, we determine the results of multiplication by powers of x:

(01010111) * (00000010) = (10101110) (01010111) * (00000100) = (01011100) ⊕ (00011011) = (01000111) (01010111) * (00001000) = (10001110) (01010111) * (00010000) = (00011100) ⊕ (00011011) = (00000111) (01010111) * (00100000) = (00001110) (01010111) * (01000000) = (00011100) (01010111) * (10000000) = (00111000)

So,

(01010111) * (10000011) = (01010111) * [(00000001) ⊕ (00000010) ⊕ (10000000)] = (01010111) ⊕ (10101110) ⊕ (00111000) = (11000001)

which is equivalent to x7 + x6 + 1.


Using a Generator

An equivalent technique for defining a finite field of the form GF(2n), using the same irreducible polynomial, is sometimes more convenient. To begin, we need two definitions: A generator g of a finite field F of order q (contains q elements) is an element whose first q - 1 powers generate all the nonzero elements of F. That is, the elements of F consist of 0, g0, g1, c, gq - 2. Consider a field F defined by a polyno-mial f(x). An element b contained in F is called a root of the polynomial if f(b) = 0. Finally, it can be shown that a root g of an irreducible polynomial is a generator of the finite field defined on that polynomial.

Table 4.9 Generator for GF(23) using x3 + x + 1

Power Representation

Polynomial Representation

Binary Representation

Decimal (Hex) Representation

0 0 000 0

g0( = g7) 1 001 1

g1 g 010 2

g2 g2 100 4

g3 g + 1 011 3

g4 g2 + g 110 6

g5 g2 + g + 1 111 7

g6 g2 + 1 101 5

Let us consider the finite field GF(23), defined over the irreducible poly-nomial x3 + x + 1, discussed previously. Thus, the generator g must satisfy f(g) = g3 + g + 1 = 0. Keep in mind, as discussed previously, that we need not find a numerical solution to this equality. Rather, we deal with polyno-mial arithmetic in which arithmetic on the coefficients is performed modulo 2. Therefore, the solution to the preceding equality is g3 = -g - 1 = g + 1. We now show that g in fact generates all of the polynomials of degree less than 3. We have the following.

g4 = g(g3) = g(g + 1) = g2 + g g5 = g(g4) = g(g2 + g) = g3 + g2 = g2 + g + 1 g6 = g(g5) = g(g2 + g + 1) = g3 + g2 + g = g2 + g + g + 1 = g2 + 1 g7 = g(g6) = g(g2 + 1) = g3 + g = g + g + 1 = 1 = g0

We see that the powers of g generate all the nonzero polynomials in GF(23). Also, it should be clear that gk = gk mod7 for any integer k. Table 4.9 shows the power representation, as well as the polynomial and binary representations.

(Continued)


This power representation makes multiplication easy. To multiply in the power notation, add exponents modulo 7. For example, g4 + g6 = g(10 mod7) = g3 = g + 1. The same result is achieved using polynomial arithmetic: We have g4 = g2 + g and g6 = g2 + 1. Then, (g2 + g) * (g2 + 1) = g4 + g3 + g2 + 1. Next, we need to determine (g4 + g3 + g2 + 1) mod (g3 + g + 1) by division:

g + 1g3 + g + 1�g4 + g3 + g2 + g

g4 + g2 + g

g3

g3 + g + 1

g + 1

We get a result of g + 1, which agrees with the result obtained using the power representation.

Table 4.10 shows the addition and multiplication tables for GF(23) using the power representation. Note that this yields the identical results to the polynomial representation (Table 4.7) with some of the rows and columns interchanged.

In general, for GF(2n) with irreducible polynomial f(x), determine gn = f(g) - gn. Then calculate all of the powers of g from gn + 1 through g2n - 2. The elements of the field correspond to the powers of g from g0 through g2n - 2 plus the value 0. For multiplication of two elements in the field, use the equality gk = gk mod(2n - 1) for any integer k.

Summary

In this section, we have shown how to construct a finite field of order 2n. Specifically, we defined GF(2n) with the following properties.

1. GF(2n) consists of 2n elements.

2. The binary operations + and * are defined over the set. The operations of addition, subtraction, multiplication, and division can be performed without leaving the set. Each element of the set other than 0 has a multi-plicative inverse.

We have shown that the elements of GF(2n) can be defined as the set of all polynomials of degree n - 1 or less with binary coefficients. Each such polynomial can be represented by a unique n-bit value. Arithmetic is defined as polynomial arithmetic modulo some irreducible polynomial of degree n. We have also seen that an equivalent definition of a finite field GF(2n) makes use of a generator and that arithmetic is defined using powers of the generator.

(Continued)

Table 4.10 GF(23) Arithmetic Using Generator for the Polynomial (x3 + x + 1)

000 001 010 100 011 110 111 101

+ 0 1 G g2 g3 g4 g5 g6

000 0 0 1 G g2 g + 1 g2 + g g2 + g + 1 g2 + 1001 1 1 0 g + 1 g2 + 1 g g2 + g + 1 g2 + g g2

010 g g g + 1 0 g2 + g 1 g2 g2 + 1 g2 + g + 1100 g2 g2 g2 + 1 g2 + g 0 g2 + g + 1 g g + 1 1

011 g3 g + 1 g 1 g2 + g + 1 0 g2 + 1 g2 g2 + g

110 g4 g2 + g g2 + g + 1 g2 g g2 + 1 0 1 g + 1

111 g5 g2 + g + 1 g2 + g g2 + 1 g + 1 g2 1 0 g

101 g6 g2 + 1 g2 g2 + g + 1 1 g2 + g g + 1 g 0

(a) Addition

000 001 010 100 011 110 111 101

* 0 1 G g2 g3 g4 g5 g6

000 0 0 0 0 0 0 0 0 0

001 1 0 1 G g2 g + 1 g2 + g g2 + g + 1 g2 + 1

010 g 0 g g2 g + 1 g2 + g g2 + g + 1 g2 + 1 1

100 g2 0 g2 g + 1 g2 + g g2 + g + 1 g2 + 1 1 g

011 g3 0 g + 1 g2 + g g2 + g + 1 g2 + 1 1 g g2

110 g4 0 g2 + g g2 + g + 1 g2 + 1 1 g g2 g + 1

111 g5 0 g2 + g + 1 g2 + 1 1 g g2 g + 1 g2 + g

101 g6 0 g2 + 1 1 g g2 g + 1 g2 + g g2 + g + 1

(b) Multiplication123


4.8 recOmmenDeD reaDing

[HERS75], still in print, is the classic treatment of abstract algebra; it is readable and rigorous. [DESK92] is another good resource. [KNUT98] provides good coverage of polynomial arithmetic.

One of the best treatments of the topics of this chapter is [BERL84], still in print. [GARR01] also has extensive coverage. A thorough and rigorous treatment of finite fields is [LIDL94]. Another solid treatment is [MURP00]. [HORO71] is a good overview of the topics of this chapter.

BERL84 Berlekamp, E. Algebraic Coding Theory. Laguna Hills, CA: Aegean Park Press, 1984.

DESK92 Deskins, W. Abstract Algebra. New York: Dover, 1992.GARR01 Garrett, P. Making, Breaking Codes: An Introduction to Cryptology. Upper

Saddle River, NJ: Prentice Hall, 2001.HERS75 Herstein, I. Topics in Algebra. New York: Wiley, 1975.HORO71 Horowitz, E. “Modular Arithmetic and Finite Field Theory: A Tutorial.”

Proceedings of the Second ACM Symposium and Symbolic and Algebraic Manipulation, March 1971.

KNUT98 Knuth, D. The Art of Computer Programming, Volume 2: Seminumerical Algorithms. Reading, MA: Addison-Wesley, 1998.

LIDL94 Lidl, R. and Niederreiter, H. Introduction to Finite Fields and Their Applications. Cambridge: Cambridge University Press, 1994.

MURP00 Murphy, T. Finite Fields. University of Dublin, Trinity College, School of Mathematics. 2000. Document available at this book’s Premium Content Web site.

4.9 Key terms, review QuestiOns, anD prObLems

Key Terms

abelian groupassociativecoefficient setcommutativecommutative ringcyclic groupdivisorEuclidean algorithmfieldfinite fieldfinite groupgenerator

greatest common divisorgroupidentity elementinfinite fieldinfinite groupinfinite ringintegral domaininverse elementirreducible polynomialmodular arithmeticmodular polynomial

arithmetic

modulusmonic polynomialorderpolynomialpolynomial arithmeticpolynomial ringprime numberprime polynomialrelatively primeresiduering

4.9 / Key terms, review Questions, and proBlems 125

Review Questions 4.1 Briefly define a group. 4.2 Briefly define a ring. 4.3 Briefly define a field. 4.4 What does it mean to say that b is a divisor of a? 4.5 What is the difference between modular arithmetic and ordinary arithmetic? 4.6 List three classes of polynomial arithmetic.

Problems 4.1 For the group Sn of all permutations of n distinct symbols, a. what is the number of elements in Sn? b. show that Sn is not abelian for n 7 2. 4.2 Does the set of residue classes (mod3) form a group a. with respect to modular addition? b. with respect to modular multiplication? 4.3 Consider the set S = {a, b} with addition and multiplication defined by the following

tables.

Is S a ring? Justify your answer. 4.4 Reformulate Equation (4.1), removing the restriction that a is a nonnegative integer.

That is, let a be any integer. 4.5 Draw a figure similar to Figure 4.1 for a 6 0. 4.6 For each of the following equations, find an integer x that satisfies the equation. a. 5x K 4 (mod 3) b. 7x K 6 (mod 5) c. 9x K 8 (mod 7) 4.7 In this text, we assume that the modulus is a positive integer. But the definition of the

expression a mod n also makes perfect sense if n is negative. Determine the following: a. 5 mod 3 b. 5 mod -3 c. -5 mod 3 d. -5 mod -3 4.8 A modulus of 0 does not fit the definition but is defined by convention as follows:

a mod 0 = a. With this definition in mind, what does the following expression mean: a K b (mod 0)?

4.9 In Section 4.3, we define the congruence relationship as follows: Two integers a and b are said to be congruent modulo n if (a mod n) = (b mod n). We then proved that a K b (mod n) if n � (a - b). Some texts on number theory use this latter relation-ship as the definition of congruence: Two integers a and b are said to be congruent modulo n if n � (a - b). Using this latter definition as the starting point, prove that, if (a mod n) = (b mod n), then n divides (a - b).

4.10 What is the smallest positive integer that has exactly k divisors, for 1 … k … 6? 4.11 Prove the following: a. a K b (mod n) implies b K a (mod n) b. a K b (mod n) and b K c (mod n) imply a K c (mod n)

+ a b

a a b

b b a

* a b

a a a

b a b


4.12 Prove the following: a. [(a mod n) - (b mod n)] mod n = (a - b) mod n b. [(a mod n) * (b mod n)] mod n = (a * b) mod n 4.13 Find the multiplicative inverse of each nonzero element in Z5. 4.14 Show that an integer N is congruent modulo 9 to the sum of its decimal digits. For

example, 475 K 4 + 7 + 5 K 16 K 1 + 6 K 7 (mod 9). This is the basis for the familiar procedure of “casting out 9’s” when checking computations in arithmetic.

4.15 a. Determine gcd(24140, 16762). b. Determine gcd(4655, 12075). 4.16 The purpose of this problem is to set an upper bound on the number of iterations of

the Euclidean algorithm. a. Suppose that m = qn + r with q 7 0 and 0 … r 6 n. Show that m/2 7 r. b. Let Ai be the value of A in the Euclidean algorithm after the ith iteration. Show that

Ai+ 2 6 Ai

2

c. Show that if m, n, and N are integers with (1 … m, n, … 2N), then the Euclidean algorithm takes at most 2N steps to find gcd(m, n).

4.17 The Euclidean algorithm has been known for over 2000 years and has always been a favorite among number theorists. After these many years, there is now a potential competitor, invented by J. Stein in 1961. Stein’s algorithms is as follows. Determine gcd(A, B) with A, B Ú 1.

STEP 1 Set A1 = A, B1 = B, C1 = 1 STEP 2 n (1) If An = Bn stop. gcd(A, B) = AnCn

(2) If An and Bn are both even, set An + 1 = An/2, Bn + 1 = Bn/2, Cn + 1 = 2Cn

(3) If An is even and Bn is odd, set An + 1 = An/2, Bn + 1 = Bn, Cn + 1 = Cn

(4) If An is odd and Bn is even, set An + 1 = An, Bn + 1 = Bn/2, Cn + 1 = Cn

(5) If An and Bn are both odd, set An + 1 = � An - Bn � , Bn + 1 =min (Bn, An), Cn + 1 = Cn

Continue to step n + 1. a. To get a feel for the two algorithms, compute gcd(2152, 764) using both the Euclid-

ean and Stein’s algorithm. b. What is the apparent advantage of Stein’s algorithm over the Euclidean algorithm? 4.18 a. Show that if Stein’s algorithm does not stop before the nth step, then

Cn + 1 * gcd(An + 1, Bn + 1) = Cn * gcd(An, Bn)

b. Show that if the algorithm does not stop before step (n - 1), then

An + 2Bn + 2 … AnBn

2

c. Show that if 1 … A, B … 2N, then Stein’s algorithm takes at most 4N steps to find gcd(m, n). Thus, Stein’s algorithm works in roughly the same number of steps as the Euclidean algorithm.

d. Demonstrate that Stein’s algorithm does indeed return gcd(A, B). 4.19 Using the extended Euclidean algorithm, find the multiplicative inverse of a. 1234 mod 4321 b. 24140 mod 40902 c. 550 mod 1769 4.20 Develop a set of tables similar to Table 4.5 for GF(5). 4.21 Demonstrate that the set of polynomials whose coefficients form a field is a ring. 4.22 Demonstrate whether each of these statements is true or false for polynomials over a field. a. The product of monic polynomials is monic.

appendiX 4a / the meaning oF mod 127

b. The product of polynomials of degrees m and n has degree m + n. c. The sum of polynomials of degrees m and n has degree max [m, n]. 4.23 For polynomial arithmetic with coefficients in Z10, perform the following

calculations. a. (7x + 2) - (x2 + 5) b. (6x2 + x + 3) * (5x2 + 2) 4.24 Determine which of the following are reducible over GF(2). a. x3 + 1 b. x3 + x2 + 1 c. x4 + 1 (be careful) 4.25 Determine the gcd of the following pairs of polynomials. a. x3 + x + 1 and x2 + x + 1 over GF(2) b. x3 - x + 1 and x2 + 1 over GF(3) c. x5 + x4 + x3 - x2 - x + 1 and x3 + x2 + x + 1 over GF(3) d. x5 + 88x4 + 73x3 + 83x2 + 51x + 67 and x3 + 97x2 + 40x + 38 over GF(101) 4.26 Develop a set of tables similar to Table 4.7 for GF(4) with m(x) = x2 + x + 1. 4.27 Determine the multiplicative inverse of x3 + x + 1 in GF(24) with m(x) = x4 + x + 1. 4.28 Develop a table similar to Table 4.9 for GF(24) with m(x) = x4 + x + 1.

Programming Problems

4.29 Write a simple four-function calculator in GF(24). You may use table lookups for the multiplicative inverses.

4.30 Write a simple four-function calculator in GF(28). You should compute the multipli-cative inverses on the fly.

appenDiX 4a the meaning OF mOD

The operator mod is used in this book and in the literature in two different ways: as a binary operator and as a congruence relation. This appendix explains the distinc-tion and precisely defines the notation used in this book regarding parentheses. This notation is common but, unfortunately, not universal.

The Binary Operator mod

If a is an integer and n is a positive integer, we define a mod n to be the remainder when a is divided by n. The integer n is called the modulus, and the remainder is called the residue. Thus, for any integer a, we can always write

a = :a/n; * n + (a mod n)

Formally, we define the operator mod as

a mod n = a - :a/n; * n for n ≠ 0

As a binary operation, mod takes two integer arguments and returns the remainder. For example, 7 mod 3 = 1. The arguments may be integers, integer


variables, or integer variable expressions. For example, all of the following are valid, with the obvious meanings:

7 mod 3

7 mod m

x mod 3

x mod m

(x2 + y + 1) mod (2m + n)

where all of the variables are integers. In each case, the left-hand term is divided by the right-hand term, and the resulting value is the remainder. Note that if either the left- or right-hand argument is an expression, the expression is parenthesized. The operator mod is not inside parentheses.

In fact, the mod operation also works if the two arguments are arbitrary real num-bers, not just integers. In this book, we are concerned only with the integer operation.

The Congruence Relation mod

As a congruence relation, mod expresses that two arguments have the same re-mainder with respect to a given modulus. For example, 7 K 4 (mod 3) expresses the fact that both 7 and 4 have a remainder of 1 when divided by 3. The following two expressions are equivalent:

a K b (mod m) 3 a mod m = b mod m

Another way of expressing it is to say that the expression a K b (mod m) is the same as saying that a - b is an integral multiple of m. Again, all the arguments may be integers, integer variables, or integer variable expressions. For example, all of the following are valid, with the obvious meanings:

7 K 4 (mod 3)

x K y (mod m)

(x2 + y + 1) K (a + 1) (mod [m + n])

where all of the variables are integers. Two conventions are used. The congruence sign is K . The modulus for the relation is defined by placing the mod operator followed by the modulus in parentheses.

The congruence relation is used to define residue classes. Those numbers that have the same remainder r when divided by m form a residue class (mod m). There are m residue classes (mod m). For a given remainder r, the residue class to which it belongs consists of the numbers

r, r { m, r { 2m, c

According to our definition, the congruence

a K b (mod m)

signifies that the numbers a and b differ by a multiple of m. Consequently, the con-gruence can also be expressed in the terms that a and b belong to the same residue class (mod m).

129

5.1 Finite Field Arithmetic

5.2 AES Structure

General StructureDetailed Structure

5.3 AES Transformation Functions

Substitute Bytes TransformationShiftRows TransformationMixColumns TransformationAddRoundKey Transformation

5.4 AES Key Expansion

Key Expansion AlgorithmRationale

5.5 An AES Example

ResultsAvalanche Effect

5.6 AES Implementation

Equivalent Inverse CipherImplementation Aspects



Appendix 5A Polynomials with Coefficients in GF(28)

Appendix 5B Simplified AES

Chapter

AdvAnced encryption StAndArd

130 Chapter 5 / advanCed enCryption Standard

“It seems very simple.”“I have solved other ciphers of an abstruseness ten thousand times greater. Circum-stances, and a certain bias of mind, have led me to take interest in such riddles, and it may well be doubted whether human ingenuity can construct an enigma of the kind which human ingenuity may not, by proper application, resolve.”

—The Gold Bug, Edgar Allen Poe

The Advanced Encryption Standard (AES) was published by the National Institute of Standards and Technology (NIST) in 2001. AES is a symmetric block cipher that is intended to replace DES as the approved standard for a wide range of applications. Compared to public-key ciphers such as RSA, the structure of AES and most symmet-ric ciphers is quite complex and cannot be explained as easily as many other crypto-graphic algorithms. Accordingly, the reader may wish to begin with a simplified version of AES, which is described in Appendix 5B. This version allows the reader to perform encryption and decryption by hand and gain a good understanding of the working of the algorithm details. Classroom experience indicates that a study of this simplified version enhances understanding of AES.1 One possible approach is to read the chapter first, then carefully read Appendix 5B, and then re-read the main body of the chapter.

Appendix H looks at the evaluation criteria used by NIST to select from among the candidates for AES, plus the rationale for picking Rijndael, which was the winning candidate. This material is useful in understanding not just the AES design but also the criteria by which to judge any symmetric encryption algorithm.

5.1 Finite Field Arithmetic

In AES, all operations are performed on 8-bit bytes. In particular, the arithmetic operations of addition, multiplication, and division are performed over the finite field GF(28). Section 4.7 discusses such operations in some detail. For the reader who has not studied Chapter 4, and as a quick review for those who have, this section summarizes the important concepts.

In essence, a field is a set in which we can do addition, subtraction, multipli-cation, and division without leaving the set. Division is defined with the following

leArning Objectives


u Present an overview of the general structure of Advanced Encryption Standard (AES).

u Understand the four transformations used in AES.u Explain the AES key expansion algorithm.u Understand the use of polynomials with coefficients in GF(28).

1However, you may safely skip Appendix 5B, at least on a first reading. If you get lost or bogged down in the details of AES, then you can go back and start with simplified AES.

5.1 / Finite Field arithmetiC 131

rule: a/b = a(b- 1). An example of a finite field (one with a finite number of elements) is the set Zp consisting of all the integers {0, 1, c, p - 1}, where p is a prime num-ber and in which arithmetic is carried out modulo p.

Virtually all encryption algorithms, both conventional and public-key, involve arithmetic operations on integers. If one of the operations used in the algorithm is division, then we need to work in arithmetic defined over a field; this is because division requires that each nonzero element have a multiplicative inverse. For con-venience and for implementation efficiency, we would also like to work with inte-gers that fit exactly into a given number of bits, with no wasted bit patterns. That is, we wish to work with integers in the range 0 through 2n - 1, which fit into an n-bit word. Unfortunately, the set of such integers, Z2n, using modular arithmetic, is not a field. For example, the integer 2 has no multiplicative inverse in Z2n, that is, there is no integer b, such that 2b mod 2n = 1.

There is a way of defining a finite field containing 2n elements; such a field is referred to as GF(2n). Consider the set, S, of all polynomials of degree n - 1 or less with binary coefficients. Thus, each polynomial has the form

f(x) = an - 1xn - 1 + an - 2x

n - 2 + g + a1x + a0 = an - 1

i = 0aix

i

where each ai takes on the value 0 or 1. There are a total of 2n different polynomials in S. For n = 3, the 23 = 8 polynomials in the set are

0 x x2 x2 + x 1 x + 1 x2 + 1 x2 + x + 1

With the appropriate definition of arithmetic operations, each such set S is a finite field. The definition consists of the following elements.

1. Arithmetic follows the ordinary rules of polynomial arithmetic using the basic rules of algebra with the following two refinements.

2. Arithmetic on the coefficients is performed modulo 2. This is the same as the XOR operation.

3. If multiplication results in a polynomial of degree greater than n - 1, then the polynomial is reduced modulo some irreducible polynomial m(x) of degree n. That is, we divide by m(x) and keep the remainder. For a polynomial f(x), the remainder is expressed as r(x) = f(x) mod m(x). A polynomial m(x) is called irreducible if and only if m(x) cannot be expressed as a product of two polyno-mials, both of degree lower than that of m(x).

For example, to construct the finite field GF(23), we need to choose an irre-ducible polynomial of degree 3. There are only two such polynomials: (x3 + x2 + 1) and (x3 + x + 1). Addition is equivalent to taking the XOR of like terms. Thus, (x + 1) + x = 1.

A polynomial in GF(2n) can be uniquely represented by its n binary coeffi-cients (an - 1an - 2 ca0). Therefore, every polynomial in GF(2n) can be represented by an n-bit number. Addition is performed by taking the bitwise XOR of the two n-bit elements. There is no simple XOR operation that will accomplish multiplication in


GF(2n). However, a reasonably straightforward, easily implemented, technique is available. In essence, it can be shown that multiplication of a number in GF(2n) by 2 consists of a left shift followed by a conditional XOR with a constant. Multiplication by larger numbers can be achieved by repeated application of this rule.

For example, AES uses arithmetic in the finite field GF(28) with the irreducible polynomial m(x) = x8 + x4 + x3 + x + 1. Consider two elements A = (a7a6 ca1a0) and B = (b7b6 cb1b0). The sum A + B = (c7c6cc1c0), where ci = ai ⊕ bi. The multiplication {02} # A equals (a6ca1a00) if a7 = 0 and equals (a6ca1a00) ⊕ (00011011) if a7 = 1.2

To summarize, AES operates on 8-bit bytes. Addition of two bytes is defined as the bitwise XOR operation. Multiplication of two bytes is defined as multiplication in the finite field GF(28), with the irreducible polynomial3 m(x) = x8 + x4 + x3 +x + 1. The developers of Rijndael give as their motivation for selecting this one of the 30 possible irreducible polynomials of degree 8 that it is the first one on the list given in [LIDL94].

5.2 Aes structure

General Structure

Figure 5.1 shows the overall structure of the AES encryption process. The cipher takes a plaintext block size of 128 bits, or 16 bytes. The key length can be 16, 24, or 32 bytes (128, 192, or 256 bits). The algorithm is referred to as AES-128, AES-192, or AES-256, depending on the key length.

The input to the encryption and decryption algorithms is a single 128-bit block. In FIPS PUB 197, this block is depicted as a 4 * 4 square matrix of bytes. This block is copied into the State array, which is modified at each stage of encryption or decryption. After the final stage, State is copied to an output matrix. These operations are depicted in Figure 5.2a. Similarly, the key is depicted as a square matrix of bytes. This key is then expanded into an array of key schedule words. Figure 5.2b shows the expansion for the 128-bit key. Each word is four bytes, and the total key schedule is 44 words for the 128-bit key. Note that the ordering of bytes within a matrix is by column. So, for example, the first four bytes of a 128-bit plaintext input to the encryp-tion cipher occupy the first column of the in matrix, the second four bytes occupy the second column, and so on. Similarly, the first four bytes of the expanded key, which form a word, occupy the first column of the w matrix.

The cipher consists of N rounds, where the number of rounds depends on the key length: 10 rounds for a 16-byte key, 12 rounds for a 24-byte key, and 14 rounds for a 32-byte key (Table 5.1). The first N - 1 rounds consist of four distinct transfor-mation functions: SubBytes, ShiftRows, MixColumns, and AddRoundKey, which are described subsequently. The final round contains only three transformations, and there is a initial single transformation (AddRoundKey) before the first round, which can be considered Round 0. Each transformation takes one or more 4 * 4 matrices

2In FIPS PUB 197, a hexadecimal number is indicated by enclosing it in curly brackets. We use that convention in this chapter.3In the remainder of this discussion, references to GF(28) refer to the finite field defined with this polynomial.

5.2 / aeS StruCture 133

Initial transformation

Key

exp

ansi

on

Plaintext—16 bytes (128 bits) Key—M bytes

Key(M bytes)Round 0 key

(16 bytes)

Round 1 key(16 bytes)

Round N – 1 key(16 bytes)

Round N key(16 bytes)

Cipehertext—16 bytes (128 bits)

No. ofrounds

10 16

KeyLength(bytes)

Input state(16 bytes)

State afterinitial

transformation(16 bytes)

Final state(16 bytes)

Round N – 1output state(16 bytes)

Round 1output state(16 bytes)

Round 1(4 transformations)

Round N – 1(4 transformations)

Round N(3 transformations)

12 24

14 32

Figure 5.1 AES Encryption Process

as input and produces a 4 * 4 matrix as output. Figure 5.1 shows that the output of each round is a 4 * 4 matrix, with the output of the final round being the ciphertext. Also, the key expansion function generates N + 1 round keys, each of which is a dis-tinct 4 * 4 matrix. Each round key serves as one of the inputs to the AddRoundKey transformation in each round.

in0 in4 in8 in12

in1 in5 in9 in13

in2 in6 in10 in14

in3 in7 in11 in15

k0

w0 w1 w2 • • • w43w42

k4 k8 k12

k1 k5 k9 k13

k2 k6 k10 k14

k3 k7 k11 k15

out0 out4 out8 out12




s0,0

s1,0

s2,0

s3,0

s0,1

s1,1

s2,1

s3,1

s0,2

s1,2

s2,2

s3,2

s0,3

s1,3

s2,3

s3,3

s0,0

s1,0

s2,0

s3,0

s0,1

s1,1

s2,1

s3,1

s0,2

s1,2

s2,2

s3,2

s0,3

s1,3

s2,3

s3,3

(a) Input, state array, and output

(b) Key and expanded key

Figure 5.2 AES Data Structures

134

5.2 / aeS StruCture 135

Table 5.1 AES Parameters

Key Size (words/bytes/bits) 4/16/128 6/24/192 8/32/256Plaintext Block Size (words/bytes/bits) 4/16/128 4/16/128 4/16/128Number of Rounds 10 12 14Round Key Size (words/bytes/bits) 4/16/128 4/16/128 4/16/128Expanded Key Size (words/bytes) 44/176 52/208 60/240

Detailed Structure

Figure 5.3 shows the AES cipher in more detail, indicating the sequence of transfor-mations in each round and showing the corresponding decryption function. As was done in Chapter 3, we show encryption proceeding down the page and decryption proceeding up the page.

Before delving into details, we can make several comments about the overall AES structure.

1. One noteworthy feature of this structure is that it is not a Feistel structure. Recall that, in the classic Feistel structure, half of the data block is used to modify the other half of the data block and then the halves are swapped. AES instead processes the entire data block as a single matrix during each round using substitutions and permutation.

2. The key that is provided as input is expanded into an array of forty-four 32-bit words, w[i]. Four distinct words (128 bits) serve as a round key for each round; these are indicated in Figure 5.3.

3. Four different stages are used, one of permutation and three of substitution:

• Substitute bytes: Uses an S-box to perform a byte-by-byte substitution of the block

• ShiftRows: A simple permutation

• MixColumns: A substitution that makes use of arithmetic over GF(28)

• AddRoundKey: A simple bitwise XOR of the current block with a portion of the expanded key

4. The structure is quite simple. For both encryption and decryption, the cipher begins with an AddRoundKey stage, followed by nine rounds that each includes all four stages, followed by a tenth round of three stages. Figure 5.4 depicts the structure of a full encryption round.

5. Only the AddRoundKey stage makes use of the key. For this reason, the cipher begins and ends with an AddRoundKey stage. Any other stage, applied at the beginning or end, is reversible without knowledge of the key and so would add no security.

6. The AddRoundKey stage is, in effect, a form of Vernam cipher and by itself would not be formidable. The other three stages together provide confu-sion, diffusion, and nonlinearity, but by themselves would provide no security because they do not use the key. We can view the cipher as alternating opera-tions of XOR encryption (AddRoundKey) of a block, followed by scrambling


Add round key

w[4, 7]

Plaintext(16 bytes)

Plaintext(16 bytes)

Substitute bytes

Expand key

Shift rows

Mix columnsRou

nd 1

Rou

nd 9

Rou

nd 1

0

Add round key

•••

Substitute bytes

Shift rows

Mix columns

Add round key

Substitute bytes

Shift rows

Add round key

Ciphertext(16 bytes)

(a) Encryption

Key(16 bytes)

Add round key

Inverse sub bytes

Inverse shift rows

Inverse mix cols

Rou

nd 1

0R

ound

9R

ound

1

Add round key

•••

Inverse sub bytes

Inverse shift rows

Inverse mix cols

Add round key

Inverse sub bytes

Inverse shift rows

Add round key

Ciphertext(16 bytes)

(b) Decryption

w[36, 39]

w[40, 43]

w[0, 3]

Figure 5.3 AES Encryption and Decryption

of the block (the other three stages), followed by XOR encryption, and so on. This scheme is both efficient and highly secure.

7. Each stage is easily reversible. For the Substitute Byte, ShiftRows, and MixColumns stages, an inverse function is used in the decryption algorithm. For the AddRoundKey stage, the inverse is achieved by XORing the same round key to the block, using the result that A ⊕ B ⊕ B = A.

8. As with most block ciphers, the decryption algorithm makes use of the expanded key in reverse order. However, the decryption algorithm is not

5.3 / aeS tranSFormation FunCtionS 137

SSubBytes

State

State

State

State

State

ShiftRows

MixColumns

AddRoundKey

S S S S S S S S S S S S S S S

M M M M

r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15

Figure 5.4 AES Encryption Round

identical to the encryption algorithm. This is a consequence of the particular structure of AES.

9. Once it is established that all four stages are reversible, it is easy to verify that decryption does recover the plaintext. Figure 5.3 lays out encryption and decryption going in opposite vertical directions. At each horizontal point (e.g., the dashed line in the figure), State is the same for both encryption and decryption.

10. The final round of both encryption and decryption consists of only three stages. Again, this is a consequence of the particular structure of AES and is required to make the cipher reversible.

5.3 Aes trAnsFOrmAtiOn FunctiOns

We now turn to a discussion of each of the four transformations used in AES. For each stage, we describe the forward (encryption) algorithm, the inverse (decryption) algorithm, and the rationale for the stage.


Substitute Bytes Transformation

Forward and Inverse TransFormaTIons The forward substitute byte transformation, called SubBytes, is a simple table lookup (Figure 5.5a). AES defines a 16 * 16 matrix of byte values, called an S-box (Table 5.2a), that con-tains a permutation of all possible 256 8-bit values. Each individual byte of State is mapped into a new byte in the following way: The leftmost 4 bits of the byte are used as a row value and the rightmost 4 bits are used as a column value. These row and column values serve as indexes into the S-box to select a unique 8-bit output value. For example, the hexadecimal value {95} references row 9, column 5 of the S-box, which contains the value {2A}. Accordingly, the value {95} is mapped into the value {2A}.

s0,0 s0,1 s0,2 s0,3

s1,0 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

s0,0 s0,1 s0,2 s0,3

s1,0 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

(b) Add round key transformation

(a) Substitute byte transformation

S-box

x

y

′ ′ ′ ′

′

′ ′′′

s1,1

s0,0

wi wi+2 wi+3

s0,2 s0,3

s1,0 s1,2 s1,3

=s2,0 s2,2 s2,3

s3,0 s3,2 s3,3

s1,1

s0,0 s0,2 s0,3

s1,0 s1,2 s1,3

s2,0 s2,2 s2,3

s3,0 s3,2 s3,3

s1,1

s0,1

s2,1

s3,1

wi+1

s0,1

s2,1

s3,1

s1,1

′′′

′ ′ ′ ′

′

′

′

′′

′ ′

′ ′ ′ ′

′ ′ ′

′′ ′

Figure 5.5 AES Byte-Level Operations


Table 5.2 AES S-Boxes

y

0 1 2 3 4 5 6 7 8 9 A B C D E F

0 63 7C 77 7B F2 6B 6F C5 30 01 67 2B FE D7 AB 76

1 CA 82 C9 7D FA 59 47 F0 AD D4 A2 AF 9C A4 72 C0

2 B7 FD 93 26 36 3F F7 CC 34 A5 E5 F1 71 D8 31 15

3 04 C7 23 C3 18 96 05 9A 07 12 80 E2 EB 27 B2 75

4 09 83 2C 1A 1B 6E 5A A0 52 3B D6 B3 29 E3 2F 84

5 53 D1 00 ED 20 FC B1 5B 6A CB BE 39 4A 4C 58 CF

6 D0 EF AA FB 43 4D 33 85 45 F9 02 7F 50 3C 9F A8

7 51 A3 40 8F 92 9D 38 F5 BC B6 DA 21 10 FF F3 D2x 8 CD 0C 13 EC 5F 97 44 17 C4 A7 7E 3D 64 5D 19 73

9 60 81 4F DC 22 2A 90 88 46 EE B8 14 DE 5E 0B DB

A E0 32 3A 0A 49 06 24 5C C2 D3 AC 62 91 95 E4 79

B E7 C8 37 6D 8D D5 4E A9 6C 56 F4 EA 65 7A AE 08

C BA 78 25 2E 1C A6 B4 C6 E8 DD 74 1F 4B BD 8B 8A

D 70 3E B5 66 48 03 F6 0E 61 35 57 B9 86 C1 1D 9E

E E1 F8 98 11 69 D9 8E 94 9B 1E 87 E9 CE 55 28 DF

F 8C A1 89 0D BF E6 42 68 41 99 2D 0F B0 54 BB 16

(a) S-box

y

0 1 2 3 4 5 6 7 8 9 A B C D E F

0 52 09 6A D5 30 36 A5 38 BF 40 A3 9E 81 F3 D7 FB

1 7C E3 39 82 9B 2F FF 87 34 8E 43 44 C4 DE E9 CB

2 54 7B 94 32 A6 C2 23 3D EE 4C 95 0B 42 FA C3 4E

3 08 2E A1 66 28 D9 24 B2 76 5B A2 49 6D 8B D1 25

4 72 F8 F6 64 86 68 98 16 D4 A4 5C CC 5D 65 B6 92

5 6C 70 48 50 FD ED B9 DA 5E 15 46 57 A7 8D 9D 84

6 90 D8 AB 00 8C BC D3 0A F7 E4 58 05 B8 B3 45 06

7 D0 2C 1E 8F CA 3F 0F 02 C1 AF BD 03 01 13 8A 6Bx 8 3A 91 11 41 4F 67 DC EA 97 F2 CF CE F0 B4 E6 73

9 96 AC 74 22 E7 AD 35 85 E2 F9 37 E8 1C 75 DF 6E

A 47 F1 1A 71 1D 29 C5 89 6F B7 62 0E AA 18 BE 1B

B FC 56 3E 4B C6 D2 79 20 9A DB C0 FE 78 CD 5A F4

C 1F DD A8 33 88 07 C7 31 B1 12 10 59 27 80 EC 5F

D 60 51 7F A9 19 B5 4A 0D 2D E5 7A 9F 93 C9 9C EF

E A0 E0 3B 4D AE 2A F5 B0 C8 EB BB 3C 83 53 99 61

F 17 2B 04 7E BA 77 D6 26 E1 69 14 63 55 21 0C 7D

(b) Inverse S-box


Here is an example of the SubBytes transformation:

b0

b1

b2

b3

b4

b5

b6

b7

=

1 0 0 0 1 1 1 1

1 1 0 0 0 1 1 1

1 1 1 0 0 0 1 1

1 1 1 1 0 0 0 1

1 1 1 1 1 0 0 0

0 1 1 1 1 1 0 0

0 0 1 1 1 1 1 0

0 0 0 1 1 1 1 1

b0

b1

b2

b3

b4

b5

b6

b7

+

1

1

0

0

0

1

1

0

Inversein GF(28)

Byte to bitcolumn vector

Bit columnvector to byte

Byte at row y,column x

initialized to yxyx

S(yx)

(a) Calculation of byte atrow y, column x of S-box

(a) Calculation of byte atrow y, column x of IS-box

Inversein GF(28)

Byte to bitcolumn vector

Bit columnvector to byte

Byte at row y,column x

initialized to yxyx

b0′

b′

b′

b ′1

2

3

b4

b5

b6

b7

=

0 0 1 0 0 1 0 1

1 0 0 1 0 0 1 0

0 1 0 0 1 0 0 1

1 0 1 0 0 1 0 0

0 1 0 1 0 0 1 0

0 0 1 0 1 0 0 1

1 0 0 1 0 1 0 0

0 1 0 0 1 0 1 0

b0

b1

b2

b3

b4

b5

b6

b7

+

1

0

1

0

0

0

0

0

IS(yx)

′

′

′

′

′

′

′

′

′

′

′

′

Figure 5.6 Constuction of S-Box and IS-Box

The S-box is constructed in the following fashion (Figure 5.6a).

EA 04 65 85

83 45 5D 96

5C 33 98 B0

F0 2D AD C5

87 F2 4D 97

EC 6E 4C 90

4A C3 46 E7

8C D8 95 A6

S


1. Initialize the S-box with the byte values in ascending sequence row by row. The first row contains {00}, {01}, {02}, c, {0F}; the second row contains {10}, {11}, etc.; and so on. Thus, the value of the byte at row y, column x is {yx}.

2. Map each byte in the S-box to its multiplicative inverse in the finite field GF(28); the value {00} is mapped to itself.

3. Consider that each byte in the S-box consists of 8 bits labeled (b7, b6, b5, b4, b3,b2, b1, b0). Apply the following transformation to each bit of each byte in the S-box:

bi′ = bi ⊕ b(i+ 4) mod 8 ⊕ b(i+ 5) mod 8 ⊕ b(i+ 6) mod 8 ⊕ b(i+ 7) mod 8 ⊕ ci (5.1)

where ci is the ith bit of byte c with the value {63}; that is, (c7c6c5c4c3c2c1c0) =(01100011). The prime (′) indicates that the variable is to be updated by the value on the right. The AES standard depicts this transformation in matrix form as follows.

H b′0b1′b2′b3′b4′b5′b6′b7′

X = H1 0 0 0 1 1 1 11 1 0 0 0 1 1 11 1 1 0 0 0 1 11 1 1 1 0 0 0 11 1 1 1 1 0 0 00 1 1 1 1 1 0 00 0 1 1 1 1 1 00 0 0 1 1 1 1 1

X Hb0

b1

b2

b3

b4

b5

b6

b7

X + H11000110

X (5.2)

Equation (5.2) has to be interpreted carefully. In ordinary matrix multiplica-tion,4 each element in the product matrix is the sum of products of the elements of one row and one column. In this case, each element in the product matrix is the bitwise XOR of products of elements of one row and one column. Furthermore, the final addition shown in Equation (5.2) is a bitwise XOR. Recall from Section 4.7 that the bitwise XOR is addition in GF(28).

As an example, consider the input value {95}. The multiplicative inverse in GF(28) is {95}- 1 = {8A}, which is 10001010 in binary. Using Equation (5.2),

H1 0 0 0 1 1 1 11 1 0 0 0 1 1 11 1 1 0 0 0 1 11 1 1 1 0 0 0 11 1 1 1 1 0 0 00 1 1 1 1 1 0 00 0 1 1 1 1 1 00 0 0 1 1 1 1 1

X H01010001

X ⊕ H11000110

X = H10010010

X ⊕ H11000110

X = H01010100

X4For a brief review of the rules of matrix and vector multiplication, refer to Appendix E.


The result is {2A}, which should appear in row {09} column {05} of the S-box. This is verified by checking Table 5.2a.

The inverse substitute byte transformation, called InvSubBytes, makes use of the inverse S-box shown in Table 5.2b. Note, for example, that the input {2A} produces the output {95}, and the input {95} to the S-box produces {2A}. The inverse S-box is constructed (Figure 5.6b) by applying the inverse of the transformation in Equation (5.1) followed by taking the multiplicative inverse in GF(28). The inverse transformation is

b′i = b(i+ 2) mod 8 ⊕ b(i+ 5) mod 8 ⊕ b(i+ 7) mod 8 ⊕ di

where byte d = {05}, or 00000101. We can depict this transformation as follows.

Hb0′b1′b2′b3′b4′b5′b6′b7′

X = H0 0 1 0 0 1 0 11 0 0 1 0 0 1 00 1 0 0 1 0 0 11 0 1 0 0 1 0 00 1 0 1 0 0 1 00 0 1 0 1 0 0 11 0 0 1 0 1 0 00 1 0 0 1 0 1 0

X Hb0

b1

b2

b3

b4

b5

b6

b7

X + H10100000

XTo see that InvSubBytes is the inverse of SubBytes, label the matrices in

SubBytes and InvSubBytes as X and Y, respectively, and the vector versions of constants c and d as C and D, respectively. For some 8-bit vector B, Equation (5.2) becomes B′ = XB ⊕ C. We need to show that Y(XB ⊕ C) ⊕ D = B. To multiply out, we must show YXB ⊕ YC ⊕ D = B. This becomes

H0 0 1 0 0 1 0 11 0 0 1 0 0 1 00 1 0 0 1 0 0 11 0 1 0 0 1 0 00 1 0 1 0 0 1 00 0 1 0 1 0 0 11 0 0 1 0 1 0 00 1 0 0 1 0 1 0

X H1 0 0 0 1 1 1 11 1 0 0 0 1 1 11 1 1 0 0 0 1 11 1 1 1 0 0 0 11 1 1 1 1 0 0 00 1 1 1 1 1 0 00 0 1 1 1 1 1 00 0 0 1 1 1 1 1

X Hb0

b1

b2

b3

b4

b5

b6

b7

X ⊕

H0 0 1 0 0 1 0 11 0 0 1 0 0 1 00 1 0 0 1 0 0 11 0 1 0 0 1 0 00 1 0 1 0 0 1 00 0 1 0 1 0 0 11 0 0 1 0 1 0 00 1 0 0 1 0 1 0

X H11000110

X ⊕ H10100000

X =


H1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1

X Hb0

b1

b2

b3

b4

b5

b6

b7

X ⊕ H10100000

X ⊕ H10100000

X = Hb0

b1

b2

b3

b4

b5

b6

b7

XWe have demonstrated that YX equals the identity matrix, and the YC = D,

so that YC ⊕ D equals the null vector.

raTIonale The S-box is designed to be resistant to known cryptanalytic attacks. Specifically, the Rijndael developers sought a design that has a low correlation between input bits and output bits and the property that the output is not a linear mathematical function of the input [DAEM01]. The nonlinearity is due to the use of the multiplicative inverse. In addition, the constant in Equation (5.1) was chosen so that the S-box has no fixed points [S@box(a) = a] and no “opposite fixed points” [S@box(a) = a], where a is the bitwise complement of a.

Of course, the S-box must be invertible, that is, IS@box[S@box(a)] = a. However, the S-box does not self-inverse in the sense that it is not true that S@box(a) = IS@box(a). For example, S@box({95}) = {2A}, but IS@box({95}) = {AD}.

ShiftRows Transformation

Forward and Inverse TransFormaTIons The forward shift row transformation, called ShiftRows, is depicted in Figure 5.7a. The first row of State is not altered. For the second row, a 1-byte circular left shift is performed. For the third row, a 2-byte circular left shift is performed. For the fourth row, a 3-byte circular left shift is per-formed. The following is an example of ShiftRows.

87 F2 4D 97

EC 6E 4C 90

4A C3 46 E7

8C D8 95 A6

87 F2 4D 97

6E 4C 90 EC

46 E7 4A C3

A6 8C D8 95

S

The inverse shift row transformation, called InvShiftRows, performs the circu -lar shifts in the opposite direction for each of the last three rows, with a 1-byte circular right shift for the second row, and so on.

raTIonale The shift row transformation is more substantial than it may first appear. This is because the State, as well as the cipher input and output, is treated as an array of four 4-byte columns. Thus, on encryption, the first 4 bytes of the plaintext are copied to the first column of State, and so on. Furthermore, as will be seen, the round key is applied to State column by column. Thus, a row shift moves an individual byte from one column to another, which is a linear


distance of a multiple of 4 bytes. Also note that the transformation ensures that the 4 bytes of one column are spread out to four different columns. Figure 5.4 illustrates the effect.

MixColumns Transformation

Forward and Inverse TransFormaTIons The forward mix column transformation, called MixColumns, operates on each column individually. Each byte of a column is mapped into a new value that is a function of all four bytes in that column. The transformation can be defined by the following matrix multiplication on State (Figure 5.7b):

D02 03 01 0101 02 03 0101 01 02 0303 01 01 02

T D s0,0 s0,1 s0,2 s0,3

s1,0 s1,1 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

T = D s′0,0 s′0,1 s′0,2 s′0,3

s′1,0 s′1,1 s′1,2 s′1,3

s′2,0 s′2,1 s′2,2 s′2,3

s′3,0 s′3,1 s′3,2 s′3,3

T (5.3)

Each element in the product matrix is the sum of products of elements of one row and one column. In this case, the individual additions and multiplications5 are

s0,0 s0,1 s0,2 s0,3

s1,0 s1,1 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

s0,0 s0,1 s0,2 s0,3

s1,0 s1,1 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

s0,0 s0,1 s0,2 s0,3

s1,0 s1,1 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

s0,0 s0,1 s0,2 s0,3

s1,1 s1,2 s1,3 s1,0

s2,2 s2,3 s2,0 s2,1

s3,3 s3,0 s3,1 s3,2

(a) Shift row transformation

(b) Mix column transformation

2 3 1 11 2 3 11 1 2 33 1 1 2

��

′ ′ ′ ′

′′′′

′ ′ ′ ′

′′′′

Figure 5.7 AES Row and Column Operations

5WefollowtheconventionofFIPSPUB197andusethesymbol•toindicatemultiplicationoverthefinite field GF(28) and ⊕ to indicate bitwise XOR, which corresponds to addition in GF(28).


performed in GF(28). The MixColumns transformation on a single column of State can be expressed as

s′0, j = (2 # s0, j) ⊕ (3 # s1, j) ⊕ s2, j ⊕ s3, j

s′1, j = s0, j ⊕ (2 # s1, j) ⊕ (3 # s2, j ⊕ s3, j (5.4)

s′2, j = s0, j ⊕ s1, j ⊕ (2 # s2, j) ⊕ (3 # s3, j)

s′3, j = (3 # s0, j) ⊕ s1, j ⊕ s2, j ⊕ (2 # s3, j)

The following is an example of MixColumns:

87 F2 4D 97

6E 4C 90 EC

46 E7 4A C3

A6 8C D8 95

47 40 A3 4C

37 D4 70 9F

94 E4 3A 42

ED A5 A6 BC

S

Let us verify the first column of this example. Recall from Section 4.7 that, in GF(28), addition is the bitwise XOR operation and that multiplication can be per-formed according to the rule established in Equation (4.14). In particular, multipli-cation of a value by x (i.e., by {02}) can be implemented as a 1-bit left shift followed by a conditional bitwise XOR with (0001 1011) if the leftmost bit of the original value (prior to the shift) is 1. Thus, to verify the MixColumns transformation on the first column, we need to show that

({02} # {87}) ⊕ ({03} # {6E}) ⊕ {46} ⊕ {A6} = {47} {87} ⊕ ({02} # {6E}) ⊕ ({03} # {46}) ⊕ {A6} = {37} {87} ⊕ {6E} ⊕ ({02} # {46}) ⊕ ({03} # {A6}) = {94} ({03} # {87}) ⊕ {6E} ⊕ {46} ⊕ ({02} # {A6}) = {ED}

For the first equation, we have {02} # {87} = (0000 1110) ⊕ (0001 1011) =(0001 0101) and {03} # {6E} = {6E} ⊕ ({02} # {6E}) = (0110 1110) ⊕ (1101 1100) =(1011 0010). Then,

{02} # {87} = 0001 0101 {03} # {6E} = 1011 0010 {46} = 0100 0110 {A6} = 1010 0110 0100 0111 = { 47}

The other equations can be similarly verified.The inverse mix column transformation, called InvMixColumns, is defined by

the following matrix multiplication: D 0E 0B 0D 09

09 0E 0B 0D0D 09 0E 0B0B 0D 09 0E

T D s0,0 s0,1 s0,2 s0,3

s1,0 s1,1 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

T = D s′0,0 s′0,1 s′0,2 s′0,3

s′1,0 s′1,1 s′1,2 s′1,3

s′2,0′ s′2,1 s′2,2 s′2,3

s′3,0 s′3,1 s′3,2 s′3,3

T (5.5)


It is not immediately clear that Equation (5.5) is the inverse of Equation (5.3). We need to showD 0E 0B 0D 09

09 0E 0B 0D0D 09 0E 0B0B 0D 09 0E

T D02 03 01 0101 02 03 0101 01 02 0303 01 01 02

T D s0,0 s0,1 s0,2 s0,3

s1,0 s1,1 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

T = D s0,0 s0,1 s0,2 s0,3

s1,0 s1,1 s1,2 s1,3

s2,0 s2,1 s2,2 s2,3

s3,0 s3,1 s3,2 s3,3

Twhich is equivalent to showingD 0E 0B 0D 09

09 0E 0B 0D0D 09 0E 0B0B 0D 09 0E

T D02 03 01 0101 02 03 0101 01 02 0303 01 01 02

T = D1 0 0 00 1 0 00 0 1 00 0 0 1

T (5.6)

That is, the inverse transformation matrix times the forward transformation matrix equals the identity matrix. To verify the first column of Equation (5.6), we need to show

({0E} # {02}) ⊕ {0B} ⊕ {0D} ⊕ ({09} # {03}) = {01} ({09} # {02}) ⊕ {0E} ⊕ {0B} ⊕ ({0D} # {03}) = {00} ({0D} # {02}) ⊕ {09} ⊕ {0E} ⊕ ({0B} # {03}) = {00} ({0B} # {02}) ⊕ {0D} ⊕ {09} ⊕ ({0E} # {03}) = {00}

For the first equation, we have {0E} # {02} = 00011100 and {09} # {03} = {09} ⊕ ({09} # {02}) = 00001001 ⊕ 00010010 = 00011011. Then

{0E} # {02} = 00011100 {0B} = 00001011 {0D} = 00001101 {09} # {03} = 00011011 00000001

The other equations can be similarly verified.The AES document describes another way of characterizing the MixColumns

transformation, which is in terms of polynomial arithmetic. In the standard, MixColumns is defined by considering each column of State to be a four-term poly-nomial with coefficients in GF(28). Each column is multiplied modulo (x4 + 1) by the fixed polynomial a(x), given by

a(x) = {03}x3 + {01}x2 + {01}x + {02} (5.7)

Appendix 5A demonstrates that multiplication of each column of State by a(x) can be written as the matrix multiplication of Equation (5.3). Similarly, it can be seen that the transformation in Equation (5.5) corresponds to treating


each column as a four-term polynomial and multiplying each column by b(x), given by

b(x) = {0B}x3 + {0D}x2 + {09}x + {0E} (5.8)

It readily can be shown that b(x) = a- 1(x) mod (x4 + 1).

raTIonale The coefficients of the matrix in Equation (5.3) are based on a linear code with maximal distance between code words, which ensures a good mixing among the bytes of each column. The mix column transformation combined with the shift row transformation ensures that after a few rounds all output bits depend on all input bits. See [DAEM99] for a discussion.

In addition, the choice of coefficients in MixColumns, which are all {01}, { 02}, or { 03}, was influenced by implementation considerations. As was discussed, multi-plication by these coefficients involves at most a shift and an XOR. The coefficients in InvMixColumns are more formidable to implement. However, encryption was deemed more important than decryption for two reasons:

1. For the CFB and OFB cipher modes (Figures 6.5 and 6.6; described in Chapter 6), only encryption is used.

2. As with any block cipher, AES can be used to construct a message authentica-tion code (Chapter 12), and for this, only encryption is used.

AddRoundKey Transformation

Forward and Inverse TransFormaTIons In the forward add round key transfor-mation, called AddRoundKey, the 128 bits of State are bitwise XORed with the 128 bits of the round key. As shown in Figure 5.5b, the operation is viewed as a colum-nwise operation between the 4 bytes of a State column and one word of the round key; it can also be viewed as a byte-level operation. The following is an example of AddRoundKey:

47 40 A3 4C

37 D4 70 9F

94 E4 3A 42

ED A5 A6 BC

AC 19 28 57

77 FA D1 5C

66 DC 29 00

F3 21 41 6A

EB 59 8B 1B

40 2E A1 C3

F2 38 13 42

1E 84 E7 D6

⊕ =

The first matrix is State, and the second matrix is the round key.The inverse add round key transformation is identical to the forward add

round key transformation, because the XOR operation is its own inverse.

raTIonale The add round key transformation is as simple as possible and affects every bit of State. The complexity of the round key expansion, plus the complexity of the other stages of AES, ensure security.

Figure 5.8 is another view of a single round of AES, emphasizing the mecha-nisms and inputs of each transformation.


SubBytes

State matrixat beginning

of round

State matrixat end

of round

MixColumns matrixRound

key

Variable inputConstant inputs

ShiftRows

MixColumns

AddRoundKey

S-box

02 03 01 01

01 02 03 01

01 01 02 03

03 01 01 02

Figure 5.8 Inputs for Single AES Round

5.4 Aes Key expAnsiOn

Key Expansion Algorithm

The AES key expansion algorithm takes as input a four-word (16-byte) key and produces a linear array of 44 words (176 bytes). This is sufficient to provide a four-word round key for the initial AddRoundKey stage and each of the 10 rounds of the cipher. The pseudocode on the next page describes the expansion.

The key is copied into the first four words of the expanded key. The remain-der of the expanded key is filled in four words at a time. Each added word w[i] depends on the immediately preceding word, w[i - 1], and the word four positions back, w[i - 4]. In three out of four cases, a simple XOR is used. For a word whose position in the w array is a multiple of 4, a more complex function is used. Figure 5.9 illustrates the generation of the expanded key, using the symbol g to represent that complex function. The function g consists of the following subfunctions.

5.4 / aeS Key expanSion 149

KeyExpansion (byte key[16], word w[44])

{

word temp

for (i = 0; i < 4; i++) w[i] = (key[4*i], key[4*i+1], key[4*i+2], key[4*i+3]);

for (i = 4; i < 44; i++)

{

temp = w[i - 1];

if (i mod 4 = 0) temp = SubWord (RotWord (temp)) ⊕ Rcon[i/4];

w[i] = w[i-4] ⊕ temp

}

}

k3

(a) Overall algorithm

(b) Function g

k7 k11 k15

k2 k6 k10 k14

k1 k5 k9 k13

k0 k4 k8 k12

w0 w1 w2 w3 g

w4 w5 w6 w7

w40 w41 w42 w43

g

B0 B1 B2 B3

w

w'

B1 B2 B3 B0

0 0 0

B1

S S

B2' ' B3

S S

B0' '

RCj

Figure 5.9 AES Key Expansion


1. RotWord performs a one-byte circular left shift on a word. This means that an input word [B0, B1, B2, B3] is transformed into [B1, B2, B3, B0].

2. SubWord performs a byte substitution on each byte of its input word, using the S-box (Table 5.2a).

3. The result of steps 1 and 2 is XORed with a round constant, Rcon[j].

The round constant is a word in which the three rightmost bytes are always 0. Thus, the effect of an XOR of a word with Rcon is to only perform an XOR on the leftmost byte of the word. The round constant is different for each round and is defined as Rcon[j] = (RC[j], 0, 0, 0), with RC[1] = 1, RC[j] = 2 # RC[j-1] and with multiplication defined over the field GF(28). The values of RC[j] in hexadecimal are

j 1 2 3 4 5 6 7 8 9 10

RC[j] 01 02 04 08 10 20 40 80 1B 36

For example, suppose that the round key for round 8 is

EA D2 73 21 B5 8D BA D2 31 2B F5 60 7F 8D 29 2F

Then the first 4 bytes (first column) of the round key for round 9 are calculated as follows:

i (decimal) tempAfter

RotWordAfter

SubWordRcon (9)

After XOR with Rcon

w[i- 4]w[i] = temp

⊕ w[i- 4]

36 7F8D292F 8D292F7F 5DA515D2 1B000000 46A515D2 EAD27321 AC7766F3

Rationale

The Rijndael developers designed the expansion key algorithm to be resistant to known cryptanalytic attacks. The inclusion of a round-dependent round constant eliminates the symmetry, or similarity, between the ways in which round keys are generated in different rounds. The specific criteria that were used are [DAEM99]

• Knowledge of a part of the cipher key or round key does not enable calcula-tion of many other round-key bits.

• An invertible transformation [i.e., knowledge of any Nk consecutive words of the expanded key enables regeneration of the entire expanded key (Nk = keysize in words)].

• Speed on a wide range of processors.

• Usage of round constants to eliminate symmetries.

• Diffusion of cipher key differences into the round keys; that is, each key bit affects many round key bits.

• Enough nonlinearity to prohibit the full determination of round key differ-ences from cipher key differences only.

• Simplicity of description.

5.5 / an aeS example 151

The authors do not quantify the first point on the preceding list, but the idea is that if you know less than Nk consecutive words of either the cipher key or one of the round keys, then it is difficult to reconstruct the remaining unknown bits. The fewer bits one knows, the more difficult it is to do the reconstruction or to deter-mine other bits in the key expansion.

5.5 An Aes exAmple

We now work through an example and consider some of its implications. Although you are not expected to duplicate the example by hand, you will find it informative to study the hex patterns that occur from one step to the next.

For this example, the plaintext is a hexadecimal palindrome. The plaintext, key, and resulting ciphertext are

Plaintext: 0123456789abcdeffedcba9876543210

Key: 0f1571c947d9e8590cb7add6af7f6798

Ciphertext: ff0b844a0853bf7c6934ab4364148fb9

Results

Table 5.3 shows the expansion of the 16-byte key into 10 round keys. As previously explained, this process is performed word by word, with each four-byte word oc-cupying one column of the word round-key matrix. The left-hand column shows

Table 5.3 Key Expansion for AES Example

Key Words Auxiliary Function

w0 = 0f 15 71 c9w1 = 47 d9 e8 59w2 = 0c b7 ad d6w3 = af 7f 67 98

RotWord (w3) = 7f 67 98 af = x1SubWord (x1) = d2 85 46 79 = y1Rcon (1) = 01 00 00 00y1 ⊕ Rcon (1) = d3 85 46 79 = z1

w4 = w0 ⊕ z1 = dc 90 37 b0w5 = w4 ⊕ w1 = 9b 49 df e9w6 = w5 ⊕ w2 = 97 fe 72 3fw7 = w6 ⊕ w3 = 38 81 15 a7

RotWord (w7) = 81 15 a7 38 = x2SubWord (x2) = 0c 59 5c 07 = y2Rcon (2) = 02 00 00 00y2 ⊕ Rcon (2) = 0e 59 5c 07 = z2

w8 = w4 ⊕ z2 = d2 c9 6b b7w9 = w8 ⊕ w5 = 49 80 b4 5ew10 = w9 ⊕ w6 = de 7e c6 61w11 = w10 ⊕ w7 = e6 ff d3 c6

RotWord (w11) = ff d3 c6 e6 = x3SubWord (x3) = 16 66 b4 83 = y3Rcon (3) = 04 00 00 00y3 ⊕ Rcon (3) = 12 66 b4 8e = z3

w12 = w8 ⊕ z3 = c0 af df 39w13 = w12 ⊕ w9 = 89 2f 6b 67w14 = w13 ⊕ w10 = 57 51 ad 06w15 = w14 ⊕ w11 = b1 ae 7e c0

RotWord (w15) = ae 7e c0 b1 = x4SubWord (x4) = e4 f3 ba c8 = y4Rcon (4) = 08 00 00 00y4 ⊕ Rcon (4) = ec f3 ba c8 = 4

(Continued)


the four round-key words generated for each round. The right-hand column shows the steps used to generate the auxiliary word used in key expansion. We begin, of course, with the key itself serving as the round key for round 0.

Next, Table 5.4 shows the progression of State through the AES encryption process. The first column shows the value of State at the start of a round. For the first row, State is just the matrix arrangement of the plaintext. The second, third, and fourth columns show the value of State for that round after the SubBytes, ShiftRows, and MixColumns transformations, respectively. The fifth column shows the round key. You can verify that these round keys equate with those shown in Table 5.3. The first column shows the value of State resulting from the bitwise XOR of State after the preceding MixColumns with the round key for the preceding round.

Avalanche Effect

If a small change in the key or plaintext were to produce a corresponding small change in the ciphertext, this might be used to effectively reduce the size of the

Key Words Auxiliary Function

w16 = w12 ⊕ z4 = 2c 5c 65 f1w17 = w16 ⊕ w13 = a5 73 0e 96w18 = w17 ⊕ w14 = f2 22 a3 90w19 = w18 ⊕ w15 = 43 8c dd 50

RotWord(w19) = 8c dd 50 43 = x5SubWord(x5) = 64 c1 53 1a = y5Rcon(5) = 10 00 00 00y5 ⊕ Rcon(5) = 74 c1 53 1a = z5

w20 = w16 ⊕ z5 = 58 9d 36 ebw21 = w20 ⊕ w17 = fd ee 38 7dw22 = w21 ⊕ w18 = 0f cc 9b edw23 = w22 ⊕ w19 = 4c 40 46 bd

RotWord (w23) = 40 46 bd 4c = x6SubWord (x6) = 09 5a 7a 29 = y6Rcon(6) = 20 00 00 00y6 ⊕ Rcon(6) = 29 5a 7a 29 = z6

w24 = w20 ⊕ z6 = 71 c7 4c c2w25 = w24 ⊕ w21 = 8c 29 74 bfw26 = w25 ⊕ w22 = 83 e5 ef 52w27 = w26 ⊕ w23 = cf a5 a9 ef

RotWord (w27) = a5 a9 ef cf = x7SubWord (x7) = 06 d3 bf 8a = y7Rcon (7) = 40 00 00 00y7 ⊕ Rcon(7) = 46 d3 df 8a = z7

w28 = w24 ⊕ z7 = 37 14 93 48w29 = w28 ⊕ w25 = bb 3d e7 f7w30 = w29 ⊕ w26 = 38 d8 08 a5w31 = w30 ⊕ w27 = f7 7d a1 4a

RotWord (w31) = 7d a1 4a f7 = x8SubWord (x8) = ff 32 d6 68 = y8Rcon (8) = 80 00 00 00y8 ⊕ Rcon(8) = 7f 32 d6 68 = z8

w32 = w28 ⊕ z8 = 48 26 45 20w33 = w32 ⊕ w29 = f3 1b a2 d7w34 = w33 ⊕ w30 = cb c3 aa 72w35 = w34 ⊕ w32 = 3c be 0b 3

RotWord (w35) = be 0b 38 3c = x9SubWord (x9) = ae 2b 07 eb = y9Rcon (9) = 1B 00 00 00y9 ⊕ Rcon (9) = b5 2b 07 eb = z9

w36 = w32 ⊕ z9 = fd 0d 42 cbw37 = w36 ⊕ w33 = 0e 16 e0 1cw38 = w37 ⊕ w34 = c5 d5 4a 6ew39 = w38 ⊕ w35 = f9 6b 41 56

RotWord (w39) = 6b 41 56 f9 = x10SubWord (x10) = 7f 83 b1 99 = y10Rcon (10) = 36 00 00 00y10 ⊕ Rcon (10) = 49 83 b1 99 = z10

w40 = w36 ⊕ z10 = b4 8e f3 52w41 = w40 ⊕ w37 = ba 98 13 4ew42 = w41 ⊕ w38 = 7f 4d 59 20w43 = w42 ⊕ w39 = 86 26 18 76

Table 5.3 Continued

5.5 / an aeS example 153

Table 5.4 AES Example

Start of Round After SubBytes After ShiftRows After MixColumns Round Key

01 89 fe 7623 ab dc 5445 cd ba 3267 ef 98 10

0f 47 0c af15 d9 b7 7f71 e8 ad 67c9 59 d6 98

0e ce f2 d936 72 6b 2b34 25 17 55ae b6 4e 88

ab 8b 89 3505 40 7f f118 3f f0 fce4 4e 2f c4

ab 8b 89 3540 7f f1 05f0 fc 18 3fc4 e4 4e 2f

b9 94 57 75e4 8e 16 5147 20 9a 3fc5 d6 f5 3b

dc 9b 97 3890 49 fe 8137 df 72 15b0 e9 3f a7

65 0f c0 4d74 c7 e8 d070 ff e8 2a75 3f ca 9c

4d 76 ba e392 c6 9b 7051 16 9b e59d 75 74 de

4d 76 ba e3c6 9b 70 929b e5 51 16de 9d 75 74

8e 22 db 12b2 f2 dc 92df 80 f7 c12d c5 1e 52

d2 49 de e6c9 80 7e ff6b b4 c6 d3b7 5e 61 c6

5c 6b 05 f47b 72 a2 6db4 34 31 129a 9b 7f 94

4a 7f 6b bf21 40 3a 3c8d 18 c7 c9b8 14 d2 22

4a 7f 6b bf40 3a 3c 21c7 c9 8d 1822 b8 14 d2

b1 c1 0b ccba f3 8b 07f9 1f 6a c31d 19 24 5c

c0 89 57 b1af 2f 51 aedf 6b ad 7e39 67 06 c0

71 48 5c 7d15 dc da a926 74 c7 bd24 7e 22 9c

a3 52 4a ff59 86 57 d3f7 92 c6 7a36 f3 93 de

a3 52 4a ff86 57 d3 59c6 7a f7 92de 36 f3 93

d4 11 fe 0f3b 44 06 73cb ab 62 3719 b7 07 ec

2c a5 f2 435c 73 22 8c65 0e a3 ddf1 96 90 50

f8 b4 0c 4c67 37 24 ffae a5 c1 eae8 21 97 bc

41 8d fe 2985 9a 36 16e4 06 78 879b fd 88 65

41 8d fe 299a 36 16 8578 87 e4 0665 9b fd 88

2a 47 c4 4883 e8 18 ba84 18 27 23eb 10 0a f3

58 fd 0f 4c9d ee cc 4036 38 9b 46eb 7d ed bd

72 ba cb 041e 06 d4 fab2 20 bc 6500 6d e7 4e

40 f4 1f f272 6f 48 2d37 b7 65 4d63 3c 94 2f

40 f4 1f f26f 48 2d 7265 4d 37 b72f 63 3c 94

7b 05 42 4a1e d0 20 4094 83 18 5294 c4 43 fb

71 8c 83 cfc7 29 e5 a54c 74 ef a9c2 bf 52 ef

0a 89 c1 85d9 f9 c5 e5d8 f7 f7 fb56 7b 11 14

67 a7 78 9735 99 a6 d961 68 68 0fb1 21 82 fa

67 a7 78 9799 a6 d9 3568 0f 61 68fa b1 21 82

ec 1a c0 800c 50 53 c73b d7 00 efb7 22 72 e0

37 bb 38 f714 3d d8 7d93 e7 08 a148 f7 a5 4a

db a1 f8 7718 6d 8b baa8 30 08 4eff d5 d7 aa

b9 32 41 f5ad 3c 3d f4c2 04 30 2f16 03 0e ac

b9 32 41 f53c 3d f4 ad30 2f c2 04ac 16 03 0e

b1 1a 44 173d 2f ec b60a 6b 2f 429f 68 f3 b1

48 f3 cb 3c26 1b c3 be45 a2 aa 0b20 d7 72 38

f9 e9 8f 2b1b 34 2f 084f c9 85 49bf bf 81 89

99 1e 73 f1af 18 15 3084 dd 97 3b08 08 0c a7

99 1e 73 f118 15 30 af97 3b 84 dda7 08 08 0c

31 30 3a c2ac 71 8c c446 65 48 eb6a 1c 31 62

fd 0e c5 f90d 16 d5 6b42 e0 4a 41cb 1c 6e 56

cc 3e ff 3ba1 67 59 af04 85 02 aaa1 00 5f 34

4b b2 16 e232 85 cb 79f2 97 77 ac32 63 cf 18

4b b2 16 e285 cb 79 3277 ac f2 9718 32 63 cf

4b 86 8a 36b1 cb 27 5afb f2 f2 afcc 5a 5b cf

b4 ba 7f 868e 98 4d 26f3 13 59 1852 4e 20 76

ff 08 69 640b 53 34 1484 bf ab 8f4a 7c 43 b9


plaintext (or key) space to be searched. What is desired is the avalanche effect, in which a small change in plaintext or key produces a large change in the ciphertext.

Using the example from Table 5.4, Table 5.5 shows the result when the eighth bit of the plaintext is changed. The second column of the table shows the value of the State matrix at the end of each round for the two plaintexts. Note that after just one round, 20 bits of the State vector differ. After two rounds, close to half the bits differ. This magnitude of difference propagates through the remaining rounds. A bit difference in approximately half the positions in the most desirable outcome. Clearly, if almost all the bits are changed, this would be logically equivalent to almost none of the bits being changed. Put another way, if we select two plaintexts at random, we would expect the two plaintexts to differ in about half of the bit positions and the two ciphertexts to also differ in about half the positions.

Table 5.6 shows the change in State matrix values when the same plaintext is used and the two keys differ in the eighth bit. That is, for the second case, the key is 0e1571c947d9e8590cb7add6af7f6798. Again, one round produces a signifi-cant change, and the magnitude of change after all subsequent rounds is roughly half the bits. Thus, based on this example, AES exhibits a very strong avalanche effect.

Table 5.5 Avalanche Effect in AES: Change in Plaintext

RoundNumber of Bits

that Differ

0123456789abcdeffedcba98765432100023456789abcdeffedcba9876543210

1

0 0e3634aece7225b6f26b174ed92b55880f3634aece7225b6f26b174ed92b5588

1

1 657470750fc7ff3fc0e8e8ca4dd02a9cc4a9ad090fc7ff3fc0e8e8ca4dd02a9c

20

2 5c7bb49a6b72349b05a2317ff46d1294fe2ae569f7ee8bb8c1f5a2bb37ef53d5

58

3 7115262448dc747e5cdac7227da9bd9cec093dfb7c45343d689017507d485e62

59

4 f867aee8b437a5210c24c1974cffeabc43efdb697244df808e8d9364ee0ae6f5

61

5 721eb200ba06206dcbd4bce704fa654e7b28a5d5ed643287e006c099bb375302

68

6 0ad9d85689f9f77bc1c5f71185e5fb143bc2d8b6798d8ac4fe36a1d891ac181a

64

7 db18a8ffa16d30d5f88b08d777ba4eaa9fb8b5452023c70280e5c4bb9e555a4b

67

8 f91b4fbfe934c9bf8f2f85812b08498920264e1126b219aef7feb3f9b2d6de40

65

9 cca104a13e678500ff59025f3bafaa34b56a0341b2290ba7dfdfbddcd8578205

61

10 ff0b844a0853bf7c6934ab4364148fb9612b89398d0600cde116227ce72433f0

58

5.6 / aeS implementation 155

Note that this avalanche effect is stronger than that for DES (Table 3.2), which requires three rounds to reach a point at which approximately half the bits are changed, both for a bit change in the plaintext and a bit change in the key.

5.6 Aes implementAtiOn

Equivalent Inverse Cipher

As was mentioned, the AES decryption cipher is not identical to the encryption cipher (Figure 5.3). That is, the sequence of transformations for decryption differs from that for encryption, although the form of the key schedules for encryption and decryption is the same. This has the disadvantage that two separate software or firmware modules are needed for applications that require both encryption and decryption. There is, however, an equivalent version of the decryption algorithm that has the same structure as the encryption algorithm. The equivalent version has the same sequence of transformations as the encryption algorithm (with transfor-mations replaced by their inverses). To achieve this equivalence, a change in key schedule is needed.

Table 5.6 Avalanche Effect in AES: Change in Key

RoundNumber of Bits

that Differ

0123456789abcdeffedcba98765432100123456789abcdeffedcba9876543210

0

0 0e3634aece7225b6f26b174ed92b55880f3634aece7225b6f26b174ed92b5588

1

1 657470750fc7ff3fc0e8e8ca4dd02a9cc5a9ad090ec7ff3fc1e8e8ca4cd02a9c

22

2 5c7bb49a6b72349b05a2317ff46d129490905fa9563356d15f3760f3b8259985

58

3 7115262448dc747e5cdac7227da9bd9c18aeb7aa794b3b66629448d575c7cebf

67

4 f867aee8b437a5210c24c1974cffeabcf81015f993c978a876ae017cb49e7eec

63

5 721eb200ba06206dcbd4bce704fa654e5955c91b4e769f3cb4a94768e98d5267

81

6 0ad9d85689f9f77bc1c5f71185e5fb14dc60a24d137662181e45b8d3726b2920

70

7 db18a8ffa16d30d5f88b08d777ba4eaafe8343b8f88bef66cab7e977d005a03c

74

8 f91b4fbfe934c9bf8f2f85812b084989da7dad581d1725c5b72fa0f9d9d1366a

67

9 cca104a13e678500ff59025f3bafaa340ccb4c66bbfd912f4b511d72996345e0

59

10 ff0b844a0853bf7c6934ab4364148fb9fc8923ee501a7d207ab670686839996b

53


Two separate changes are needed to bring the decryption structure in line with the encryption structure. As illustrated in Figure 5.3, an encryption round has the structure SubBytes, ShiftRows, MixColumns, AddRoundKey. The standard decryption round has the structure InvShiftRows, InvSubBytes, AddRoundKey, InvMixColumns. Thus, the first two stages of the decryption round need to be interchanged, and the second two stages of the decryption round need to be interchanged.

InTerchangIng InvshIFTrows and InvsubbyTes InvShiftRows affects the sequence of bytes in State but does not alter byte contents and does not depend on byte contents to perform its transformation. InvSubBytes affects the contents of bytes in State but does not alter byte sequence and does not depend on byte sequence to perform its transformation. Thus, these two operations commute and can be interchanged. For a given State Si,

InvShiftRows [InvSubBytes (Si)] = InvSubBytes [InvShiftRows (Si)]

InTerchangIng addroundKey and InvmIxcolumns The transformations AddRoundKey and InvMixColumns do not alter the sequence of bytes in State. If we view the key as a sequence of words, then both AddRoundKey and InvMixColumns operate on State one column at a time. These two operations are linear with respect to the column input. That is, for a given State Si and a given round key wj,

InvMixColumns (Si ⊕ wj) = [InvMixColumns (Si)] ⊕ [InvMixColumns (wj)]

To see this, suppose that the first column of State Si is the sequence (y0, y1, y2, y3) and the first column of the round key wj is (k0, k1, k2, k3). Then we need to showD 0E 0B 0D 09

09 0E 0B 0D0D 09 0E 0B0B 0D 09 0E

T Dy0 ⊕ k0

y1 ⊕ k1

y2 ⊕ k2

y3 ⊕ k3

T = D 0E 0B 0D 0909 0E 0B 0D0D 09 0E 0B0B 0D 09 0E

T Dy0

y1

y2

y3

T ⊕ D 0E 0B 0D 0909 0E 0B 0D0D 09 0E 0B0B 0D 09 0E

T Dk0

k1

k2

k3

TLet us demonstrate that for the first column entry. We need to show

[{0E} # (y0 ⊕ k0)] ⊕ [{0B} # (y1 ⊕ k1)] ⊕ [{0D} # (y2 ⊕ k2)] ⊕ [{09} # (y3 ⊕ k3)] = [{0E} # y0] ⊕ [{0B} # y1] ⊕ [{0D} # y2] ⊕ [{09} # y3] ⊕ [{0E} # k0] ⊕ [{0B} # k1] ⊕ [{0D} # k2] ⊕ [{09} # k3]

This equation is valid by inspection. Thus, we can interchange AddRoundKey and InvMixColumns, provided that we first apply InvMixColumns to the round key. Note that we do not need to apply InvMixColumns to the round key for the input to the first AddRoundKey transformation (preceding the first round) nor to the last AddRoundKey transformation (in round 10). This is because these two AddRoundKey transformations are not interchanged with InvMixColumns to pro-duce the equivalent decryption algorithm.

Figure 5.10 illustrates the equivalent decryption algorithm.

5.6 / aeS implementation 157

Implementation Aspects

The Rijndael proposal [DAEM99] provides some suggestions for efficient implemen-tation on 8-bit processors, typical for current smart cards, and on 32-bit processors, typical for PCs.

8-bIT Processor AES can be implemented very efficiently on an 8-bit proces-sor. AddRoundKey is a bytewise XOR operation. ShiftRows is a simple byte-shifting operation. SubBytes operates at the byte level and only requires a table of 256 bytes.

The transformation MixColumns requires matrix multiplication in the field GF(28), which means that all operations are carried out on bytes. MixColumns only requires multiplication by {02} and {03}, which, as we have seen, involved simple

Add round key

w[36, 39]

w[40, 43]

Ciphertext

Inverse sub bytes

Inverse shift rows

Inverse mix cols Rou

nd 1

Rou

nd 9

Rou

nd 1

0

Add round keyInverse mix cols

•••

Inverse sub bytes

Inverse shift rows

Inverse mix cols

Add round keyInverse mix cols

Inverse sub bytes

Inverse shift rowsExpand key

Add round key

PlaintextKey

w[4, 7]

w[0, 3]

Figure 5.10 Equivalent Inverse Cipher


shifts, conditional XORs, and XORs. This can be implemented in a more efficient way that eliminates the shifts and conditional XORs. Equation set (5.4) shows the equations for the MixColumns transformation on a single column. Using the iden-tity {03} # x = ({02} # x) ⊕ x, we can rewrite Equation set (5.4) as follows.

Tmp = s0, j ⊕ s1, j ⊕ s2, j ⊕ s3, j

s0, j= = s0, j ⊕ Tmp ⊕ [2 # (s0, j ⊕ s1, j)]

s1, j= = s1, j ⊕ Tmp ⊕ [2 # (s1, j ⊕ s2, j)] (5.9)

s2, j= = s2, j ⊕ Tmp ⊕ [2 # (s2, j ⊕ s3, j)]

s3, j= = s3, j ⊕ Tmp ⊕ [2 # (s3, j ⊕ s0, j)]

Equation set (5.9) is verified by expanding and eliminating terms.The multiplication by {02} involves a shift and a conditional XOR. Such an im-

plementation may be vulnerable to a timing attack of the sort described in Section 3.4. To counter this attack and to increase processing efficiency at the cost of some stor-age, the multiplication can be replaced by a table lookup. Define the 256-byte table X2, such that X2[i] = {02} # i. Then Equation set (5.9) can be rewritten as

Tmp = s0, j ⊕ s1, j ⊕ s2, j ⊕ s3, j

s0, j= = s0, j ⊕ Tmp ⊕ X2[s0, j ⊕ s1, j]

s1, c= = s1, j ⊕ Tmp ⊕ X2[s1, j ⊕ s2, j]

s2, c= = s2, j ⊕ Tmp ⊕ X2[s2, j ⊕ s3, j]

s3, j= = s3, j ⊕ Tmp ⊕ X2[s3, j ⊕ s0, j]

32-bIT Processor The implementation described in the preceding subsection uses only 8-bit operations. For a 32-bit processor, a more efficient implementation can be achieved if operations are defined on 32-bit words. To show this, we first define the four transformations of a round in algebraic form. Suppose we begin with a State matrix consisting of elements ai, j and a round-key matrix consisting of ele-ments ki, j. Then the transformations can be expressed as follows.

SubBytes bi, j = S[ai, j]

ShiftRows D c0, j

c1, j

c2, j

c3, j

T = D b0, j

b1, j- 1

b2, j- 2

b3, j- 3

TMixColumns Dd0, j

d1, j

d2, j

d3, j

T = D02 03 01 0101 02 03 0101 01 02 0303 01 01 02

T D c0, j

c1, j

c2, j

c3, j

TAddRoundKey D e0, j

e1, j

e2, j

e3, j

T = Dd0, j

d1, j

d2, j

d3, j

T ⊕ Dk0, j

k1, j

k2, j

k3, j

T

5.7 / reCommended reading 159

In the ShiftRows equation, the column indices are taken mod 4. We can com-bine all of these expressions into a single equation:

D e0, j

e1, j

e2, j

e3, j

T = D02 03 01 0101 02 03 0101 01 02 0303 01 01 02

T D S[a0, j]S[a1, j- 1]S[a2, j- 2]S[a3, j- 3]

T ⊕ Dk0, j

k1, j

k2, j

k3, j

T = § D02

010103

T # S[a0, j]¥ ⊕ § D03020101

T # S[a1, j-1]¥ ⊕ § D01030201

T # S[a2, j-2]¥ ⊕ § D01

010302

T # S[a3, j-3]¥ ⊕ Dk0, j

k1, j

k2, j

k3, j

TIn the second equation, we are expressing the matrix multiplication as a linear com-bination of vectors. We define four 256-word (1024-byte) tables as follows.

T0[x] = § D02010103

T # S[x]¥ T1[x] = § D03020101

T # S[x]¥ T2[x] = § D01030201

T # S[x]¥ T3[x] = § D01010302

T # S[x]¥Thus, each table takes as input a byte value and produces a column vector (a 32-bit word) that is a function of the S-box entry for that byte value. These tables can be calculated in advance.

We can define a round function operating on a column in the following fashion.D s′0, j

s′1, j

s′2, j

s′3, j

T = T0[s0, j] ⊕ T1[s1, j- 1] ⊕ T2[s2, j- 2] ⊕ T3[s3, j- 3] ⊕ Dk0, j

k1, j

k2, j

k3, j

TAs a result, an implementation based on the preceding equation requires only

four table lookups and four XORs per column per round, plus 4 Kbytes to store the table. The developers of Rijndael believe that this compact, efficient implementation was probably one of the most important factors in the selection of Rijndael for AES.

5.7 recOmmended reAding

The most thorough description of AES so far available is the book by the developers of AES, [DAEM02]. The authors also provide a brief description and design rationale in [DAEM01]. [LAND04] is a rigorous mathematical treatment of AES and its cryptanalysis.


Another worked-out example of AES operation, authored by instructors at Massey U., New Zealand, is available at this book’s Premium Content Web site.

DAEM01 Daemen, J., and Rijmen, V. “Rijndael: The Advanced Encryption Standard.” Dr. Dobb’s Journal, March 2001.

DAEM02 Daemen, J., and Rijmen, V. The Design of Rijndael: The Wide Trail Strategy Explained. New York: Springer-Verlag, 2002.

LAND04 Landau, S. “Polynomials in the Nation’s Service: Using Algebra to Design the Advanced Encryption Standard.” American Mathematical Monthly, February 2004.

Review Questions 5.1 What was the original set of criteria used by NIST to evaluate candidate AES ciphers? 5.2 What was the final set of criteria used by NIST to evaluate candidate AES ciphers? 5.3 What is the difference between Rijndael and AES? 5.4 What is the purpose of the State array? 5.5 How is the S-box constructed? 5.6 Briefly describe SubBytes. 5.7 Briefly describe ShiftRows. 5.8 How many bytes in State are affected by ShiftRows? 5.9 Briefly describe MixColumns. 5.10 Briefly describe AddRoundKey. 5.11 Briefly describe the key expansion algorithm. 5.12 What is the difference between SubBytes and SubWord? 5.13 What is the difference between ShiftRows and RotWord? 5.14 What is the difference between the AES decryption algorithm and the equivalent

inverse cipher?

Problems 5.1 In the discussion of MixColumns and InvMixColumns, it was stated that

b(x) = a- 1(x)mod (x4 + 1)

where a(x) = {03}x3 + {01}x2 + {01}x + {02} and b(x) = {0B}x3 + {0D}x2 + {09}x + {0E}. Show that this is true.

5.2 a. What is {01}- 1 in GF(28)? b. Verify the entry for {01} in the S-box.

5.8 Key terms, review QuestiOns, And prOblems

Key Terms

Advanced Encryption Standard (AES)

avalanche effectfield

finite fieldirreducible

polynomialkey expansion

National Institute of Standards and Technology (NIST)

RijndaelS-box

5.8 / Key termS, review QueStionS, and problemS 161

5.3 Show the first eight words of the key expansion for a 128-bit key of all zeros. 5.4 Given the plaintext {000102030405060708090A0B0C0D0E0F} and the key

{01010101010101010101010101010101}: a. Show the original contents of State, displayed as a 4 * 4 matrix. b. Show the value of State after initial AddRoundKey. c. Show the value of State after SubBytes. d. Show the value of State after ShiftRows. e. Show the value of State after MixColumns. 5.5 Verify Equation (5.11). That is, show that xi mod (x4 + 1) = xi mod 4. 5.6 Compare AES to DES. For each of the following elements of DES, indicate the com-

parable element in AES or explain why it is not needed in AES. a. XOR of subkey material with the input to the f function b. XOR of the f function output with the left half of the block c. f function d. permutation P e. swapping of halves of the block 5.7 In the subsection on implementation aspects, it is mentioned that the use of tables

helps thwart timing attacks. Suggest an alternative technique. 5.8 In the subsection on implementation aspects, a single algebraic equation is developed

that describes the four stages of a typical round of the encryption algorithm. Provide the equivalent equation for the tenth round.

5.9 Compute the output of the MixColumns transformation for the following sequence of input bytes “67 89 AB CD.” Apply the InvMixColumns transformation to the obtained result to verify your calculations. Change the first byte of the input from “67” to “77” perform the MixColumns transformation again for the new input, and determine how many bits have changed in the output.Note: You can perform all calculations by hand or write a program support-ing these computations. If you choose to write a program, it should be written entirely by you; no use of libraries or public domain source code is allowed in this assignment.

5.10 Use the key 1010 0111 0011 1011 to encrypt the plaintext “ok” as expressed in ASCII as 0110 1111 0110 1011. The designers of S-AES got the ciphertext 0000 0111 0011 1000. Do you?

5.11 Show that the matrix given here, with entries in GF(24), is the inverse of the matrix used in the MixColumns step of S-AES.

ax3 + 1 xx x3 + 1

b

5.12 Carefully write up a complete decryption of the ciphertext 0000 0111 0011 1000 using the key 1010 0111 0011 1011 and the S-AES algorithm. You should get the plaintext we started with in Problem 5.10. Note that the inverse of the S-boxes can be done with a reverse table lookup. The inverse of the MixColumns step is given by the ma-trix in the previous problem.

5.13 Demonstrate that Equation (5.9) is equivalent to Equation (5.4).

Programming Problems 5.14 Create software that can encrypt and decrypt using S-AES. Test data: A binary

plaintext of 0110 1111 0110 1011 encrypted with a binary key of 1010 0111 0011 1011 should give a binary ciphertext of 0000 0111 0011 1000. Decryption should work correspondingly.

5.15 Implement a differential cryptanalysis attack on 1-round S-AES.


Appendix 5A pOlynOmiAls with cOeFFicients in gF(28)

In Section 4.5, we discussed polynomial arithmetic in which the coefficients are in Zp and the polynomials are defined modulo a polynomial M(x) whose highest power is some integer n. In this case, addition and multiplication of coefficients occurred within the field Zp; that is, addition and multiplication were performed modulo p.

The AES document defines polynomial arithmetic for polynomials of degree 3 or less with coefficients in GF(28). The following rules apply.

1. Addition is performed by adding corresponding coefficients in GF(28). As was pointed out Section 4.5, if we treat the elements of GF(28) as 8-bit strings, then addition is equivalent to the XOR operation. So, if we have

a(x) = a3x3 + a2x

2 + a1x + a0 (5.10)

and

b(x) = b3x3 + b2x

2 + b1x + b0 (5.11)

then

a(x) + b(x) = (a3 ⊕ b3)x3 + (a2 ⊕ b2)x2 + (a1 ⊕ b1)x + (a0 ⊕ b0)

2. Multiplication is performed as in ordinary polynomial multiplication with two refinements:

a. Coefficients are multiplied in GF(28). b. The resulting polynomial is reduced mod (x4 + 1).

We need to keep straight which polynomial we are talking about. Recall from Section 4.6 that each element of GF(28) is a polynomial of degree 7 or less with binary coefficients, and multiplication is carried out modulo a polynomial of degree 8. Equivalently, each element of GF(28) can be viewed as an 8-bit byte whose bit val-ues correspond to the binary coefficients of the corresponding polynomial. For the sets defined in this section, we are defining a polynomial ring in which each ele-ment of this ring is a polynomial of degree 3 or less with coefficients in GF(28), and multiplication is carried out modulo a polynomial of degree 4. Equivalently, each element of this ring can be viewed as a 4-byte word whose byte values are elements of GF(28) that correspond to the 8-bit coefficients of the corresponding polynomial.

We denote the modular product of a(x) and b(x) by a(x) ⊗ b(x). To com-pute d(x) = a(x) ⊗ b(x), the first step is to perform a multiplication without the modulo operation and to collect coefficients of like powers. Let us express this as c(x) = a(x) * b(x). Then

c(x) = c6x6 + c5x

5 + c4x4 + c3x

3 + c2x2 + c1x + c0 (5.12)

where

c0 = a0# b0 c4 = (a3

# b1) ⊕ (a2# b2) ⊕ (a1

# b3) c1 = (a1

# b0) ⊕ (a0# b1) c5 = (a3

# b2) ⊕ (a2# b3)

c2 = (a2# b0) ⊕ (a1

# b1) ⊕ (a0# b2) c6 = a3

# b3 c3 = (a3

# b0) ⊕ (a2# b1) ⊕ (a1

# b2) ⊕ (a0# b3)

appendix 5a / polynomialS with CoeFFiCientS in gF(28) 163

The final step is to perform the modulo operation

d(x) = c(x) mod (x4 + 1)

That is, d(x) must satisfy the equation

c(x) = [(x4 + 1) * q(x)] ⊕ d(x)

such that the degree of d(x) is 3 or less.A practical technique for performing multiplication over this polynomial ring

is based on the observation that

xi mod (x4 + 1) = xi mod 4 (5.13)

If we now combine Equations (5.12) and (5.13), we end up with

d(x) = c(x) mod (x4 + 1)

= [c6x6 + c5x

5 + c4x4 + c3x

3 + c2x2 + c1x + c0] mod (x4 + 1)

= c3x3 + (c2 ⊕ c6)x2 + (c1 ⊕ c5)x + (c0 ⊕ c4)

Expanding the ci coefficients, we have the following equations for the coef-ficients of d(x).

d0 = (a0# b0) ⊕ (a3

# b1) ⊕ (a2# b2) ⊕ (a1

# b3)

d1 = (a1# b0) ⊕ (a0

# b1) ⊕ (a3# b2) ⊕ (a2

# b3)

d2 = (a2# b0) ⊕ (a1

# b1) ⊕ (a0# b2) ⊕ (a3

# b3)

d3 = (a3# b0) ⊕ (a2

# b1) ⊕ (a1# b2) ⊕ (a0

# b3)

This can be written in matrix form:

Dd0

d1

d2

d3

T = Da0 a3 a2 a1

a1 a0 a3 a2

a2 a1 a0 a3

a3 a2 a1 a0

T Db0

b1

b2

b3

T (5.14)

MixColumns Transformation

In the discussion of MixColumns, it was stated that there were two equivalent ways of defining the transformation. The first is the matrix multiplication shown in Equation (5.3), which is repeated here:D02 03 01 01

01 02 03 0101 01 02 0303 01 01 02

T D s0, 0 s0, 1 s0, 2 s0, 3

s1, 0 s1, 1 s1, 2 s1, 3

s2, 0 s2, 1 s2, 2 s2, 3

s3, 0 s3, 1 s3, 2 s3, 3

T = D s′0, 0 s′0, 1 s′0, 2 s′0, 3

s′1, 0 s′1, 1 s′1, 2 s′1, 3

s′2, 0 s′2, 1 s′2, 2 s′2, 3

s′3, 0 s′3, 1 s′3, 2 s′3, 3

TThe second method is to treat each column of State as a four-term polynomial

with coefficients in GF(28). Each column is multiplied modulo (x4 + 1) by the fixed polynomial a(x), given by

a(x) = {03}x3 + {01}x2 + {01}x + {02}


From Equation (5.10), we have a3 = {03}; a2 = {01}; a1 = {01}; and a0 = {02}. For the jth column of State, we have the polynomial colj(x) = s3, jx

3 +s2, jx

2 + s1, j x + s0, j. Substituting into Equation (5.14), we can express d(x) = a(x) * colj(x) asDd0

d1

d2

d3

T = Da0 a3 a2 a1

a1 a0 a3 a2

a2 a1 a0 a3

a3 a2 a1 a0

T D s0, j

s1, j

s2, j

s3, j

T = D02 03 01 0101 02 03 0101 01 02 0303 01 01 02

T D s0, j

s1, j

s2, j

s3, j

Twhich is equivalent to Equation (5.3).

Multiplication by x

Consider the multiplication of a polynomial in the ring by x: c(x) = x ⊕ b(x). We have

c(x) = x ⊗ b(x) = [x * (b3x3 + b2x

2 + b1x + b0] mod (x4 + 1) = (b3x

4 + b2x3 + b1x

2 + b0x) mod (x4 + 1) = b2x

3 + b1x2 + b0x + b3

Thus, multiplication by x corresponds to a 1-byte circular left shift of the 4 bytes in the word representing the polynomial. If we represent the polynomial as a 4-byte column vector, then we haveD c0

c1

c2

c3

T = D00 00 00 0101 00 00 0000 01 00 0000 00 01 00

T Db0

b1

b2

b3

T Appendix 5b simpliFied Aes

Simplified AES (S-AES) was developed by Professor Edward Schaefer of Santa Clara University and several of his students [MUSA03]. It is an educational rather than a secure encryption algorithm. It has similar properties and structure to AES with much smaller parameters. The reader might find it useful to work through an example by hand while following the discussion in this appendix. A good grasp of S-AES will make it easier for the student to appreciate the structure and workings of AES.

Overview

Figure 5.11 illustrates the overall structure of S-AES. The encryption algorithm takes a 16-bit block of plaintext as input and a 16-bit key and produces a 16-bit block of ciphertext as output. The S-AES decryption algorithm takes an 16-bit block of ciphertext and the same 16-bit key used to produce that ciphertext as input and produces the original 16-bit block of plaintext as output.

appendix 5b / SimpliFied aeS 165

The encryption algorithm involves the use of four different functions, or trans-formations: add key (AK), nibble substitution (NS), shift row (SR), and mix column (MC), whose operation is explained subsequently.

We can concisely express the encryption algorithm as a composition6 of functions:

AK2∘ SR ∘ NS ∘ AK1

∘ MC ∘ SR ∘ NS ∘ AK0

so that AK0 is applied first.

The encryption algorithm is organized into three rounds. Round 0 is simply an add key round; round 1 is a full round of four functions; and round 2 contains only 3 functions. Each round includes the add key function, which makes use of 16 bits of key. The initial 16-bit key is expanded to 48 bits, so that each round uses a distinct 16-bit round key.

Each function operates on a 16-bit state, treated as a 2 * 2 matrix of nib-bles, where one nibble equals 4 bits. The initial value of the State matrix is the 16-bit plaintext; State is modified by each subsequent function in the encryption pro-cess, producing after the last function the 16-bit ciphertext. As Figure 5.12a shows, the ordering of nibbles within the matrix is by column. So, for example, the first 8 bits of a 16-bit plaintext input to the encryption cipher occupy the first column of the matrix, and the second 8 bits occupy the second column. The 16-bit key is

Add round key

w[2, 3]

w[0, 1]

16-bit plaintext

Nibble substitution Expand key

Shift row

Mix columnsRou

nd 1

Rou

nd 2

Add round key

Nibble substitution

Shift row

Add round key

16-bit ciphertext

16-bit key

Add round key

16-bit plaintext

ENCRYPTION DECRYPTION

Inverse nibble sub

Inverse shift row

Inverse mix cols

Rou

nd 2

Rou

nd 1Add round key

Inverse nibble sub

Inverse shift row

Add round key

16-bit ciphertext

w[4, 5]

Figure 5.11 S-AES Encryption and Decryption

6Definition: If f and g are two functions, then the function F with the equation y = F(x) = g[f(x)] is called the composition of f and g and is denoted as F = g ∘ f.


similarly organized, but it is somewhat more convenient to view the key as two bytes rather than four nibbles (Figure 5.12b). The expanded key of 48 bits is treated as three round keys, whose bits are labeled as follows: K0 = k0 . . . k15; K1 = k16 . . . k31; and K2 = k32 . . . k47.

Figure 5.13 shows the essential elements of a full round of S-AES.Decryption is also shown in Figure 5.11 and is essentially the reverse of

encryption:

AK0∘ INS ∘ ISR ∘ IMC ∘ AK1

∘ INS ∘ ISR ∘ AK2

in which three of the functions have a corresponding inverse function: inverse nib-ble substitution (INS), inverse shift row (ISR), and inverse mix column (IMC).

S-AES Encryption and Decryption

We now look at the individual functions that are part of the encryption algorithm.

add Key The add key function consists of the bitwise XOR of the 16-bit State matrix and the 16-bit round key. Figure 5.14 depicts this as a columnwise operation, but it can also be viewed as a nibble-wise or bitwise operation. The following is an example.

S1,0 S1,1

S0,0 S0,1b0b1b2b3 b8b9b10b11

b4b5b6b7 b12b13b14b15

Bit representation

Bit representation

Byte representation

k0k1k2k3k4k5k6k7 k8k9k10k11k12k13k14k15

Nibble representation

Original key Key expansion

(a) State matrix

(b) Key

w0

K0

w1 w2 w3 w4 w5

K1 K2

Figure 5.12 S-AES Data Structures

2 5

D 5

8 1

A C

A 4

7 9 ⊕ =

State matrix Key

The inverse of the add key function is identical to the add key function, because the XOR operation is its own inverse.

SM

r0

State State State State State

Nibblesubstitution

Shiftrow

Mixcolumn

Add key

S

S

SM

r1

r2

r3

S0,0

S1,0

S0,1

S1,1

S0,0

S1,0

S0,1

S1,1

S0,0

S1,0

S0,1

S1,1

S0,0

S1,0

S0,1

S1,1

S0,0

S1,0

S0,1

S1,1

Figure 5.13 S-AES Encryption Round

167


s0,0 s0,1

s1,0

s0,0 s0,1

s1,0

x Nibblesubstitution

Shiftrow

Mixcolumn

Addkey

' '

's1,1 's1,1

s0,0 s0,1

s1,0 s1,1

s0,0 s0,1

s1,1 s1,0

s0,0 s0,1

s1,0 s1,1

1 44 1 ��

s0,0 s0,1

s1,0 s1,1

' '

' '

wi wi+1 �

s0,0

s1,1

s0,1

s1,0

s0,0 s0,1

s1,0

'

s1,1'

'

'

Figure 5.14 S-AES Transformations

nIbble subsTITuTIon The nibble substitution function is a simple table lookup (Figure 5.14). AES defines a 4 * 4 matrix of nibble values, called an S-box (Table 5.7a), that contains a permutation of all possible 4-bit values. Each individ-ual nibble of State is mapped into a new nibble in the following way: The leftmost 2 bits of the nibble are used as a row value, and the rightmost 2 bits are used as a column value. These row and column values serve as indexes into the S-box to select a unique 4-bit output value. For example, the hexadecimal value A references row 2, column 2 of the S-box, which contains the value 0. Accordingly, the value A is mapped into the value 0.

Here is an example of the nibble substitution transformation.

8 1

A C

6 4

C 0S

The inverse nibble substitution function makes use of the inverse S-box shown in Table 5.7b. Note, for example, that the input 0 produces the output A, and the input A to the S-box produces 0.


shIFT row The shift row function performs a one-nibble circular shift of the sec-ond row of State the first row is not altered (Figure 5.14). The following is an example.

Table 5.7 S-AES S-Boxes

j

00 01 10 11

i

00 9 4 A B

01 D 1 8 5

10 6 2 0 3

11 C E F 7

(a) S-Box

j

00 01 10 11

i

00 A 5 9 B

01 1 7 8 F

10 6 0 2 3

11 C 4 D E

(b) Inverse S-Box

Note: Hexadecimal numbers in shaded boxes; binary numbers in unshaded boxes.

6 4

0 C

6 4

C 0S

The inverse shift row function is identical to the shift row function, because it shifts the second row back to its original position.

mIx column The mix column function operates on each column individually. Each nibble of a column is mapped into a new value that is a function of both nibbles in that column. The transformation can be defined by the following matrix multiplica-tion on State (Figure 5.14):

c1 44 1

d c s0,0 s0,1

s1,0 s1,1d = c s′0, 0 s′0, 1

s′1,0 s′1,1d

Performing the matrix multiplication, we get

S0,0= = S0,0 ⊕ (4 # S1,0)

S1,0= = (4 # S0,0) ⊕ S1,0

S0,1= = S0,1 ⊕ (4 # S1,1)

S1,1= = (4 # S01,) ⊕ S1,1

Where arithmetic is performed in GF(24),andthesymbol•referstomultiplicationin GF(24). Appendix I provides the addition and multiplication tables. The follow-ing is an example.

c1 44 1

d c 6 4C 0

d = c3 47 3

d

The inverse mix column function is defined as

c9 22 9

d c s0, 0 s0,1

s1,0 s1,1d = c s′0, 0 s′0,1

s′1, 0 s′1, 1d


We demonstrate that we have indeed defined the inverse in the following fashion.

c9 22 9

d c1 44 1

d c s0,0 s0,1

s1,0 s1,1d = c1 0

0 1d c s0,0 s0,1

s1,0 s1,1d = c s0,0 s0,1

s1,0 s1,1d

The preceding matrix multiplication makes use of the following results in GF(24): 9 + (2 # 4) = 9 + 8 = 1 and (9 # 4) + 2 = 2 + 2 = 0. These opera-tions can be verified using the arithmetic tables in Appendix I or by polynomial arithmetic.

The mix column function is the most difficult to visualize. Accordingly, we provide an additional perspective on it in Appendix I.

Key exPansIon For key expansion, the 16 bits of the initial key are grouped into a row of two 8-bit words. Figure 5.15 shows the expansion into six words, by the calcu-lation of four new words from the initial two words. The algorithm is

(b) Function g

g

N0 N1

w

w'

N1 N0

xi+2 0

N 1

S S

' N 0'

(a) Overall algorithm

w0 w1

g

w2 w3 g

w4 w5

Figure 5.15 S-AES Key Expansion


w2 = w0 ⊕ g(w1) = w0 ⊕ Rcon(1) ⊕ SubNib(RotNib(w1))

w3 = w2 ⊕ w1

w4 = w2 ⊕ g(w3) = w2 ⊕ Rcon(2) ⊕ SubNib(RotNib(w3))

w5 = w4 ⊕ w3

Rcon is a round constant, defined as follows: RC[i] = xi+ 2, so that RC[1] = x3 = 1000 and RC[2] = x4mod (x4 + x + 1) = x + 1 = 0011. RC[i] forms the leftmost nibble of a byte, with the rightmost nibble being all zeros. Thus, Rcon(1) = 10000000 and Rcon(2) = 00110000.

For example, suppose the key is 2D55 = 0010 1101 0101 0101 = w0w1. Then

w2 = 00101101 ⊕ 10000000 ⊕ SubNib(01010101) = 00101101 ⊕ 10000000 ⊕ 00010001 = 10111100

w3 = 10111100 ⊕ 01010101 = 11101001 w4 = 10111100 ⊕ 00110000 ⊕ SubNib(10011110)

= 10111100 ⊕ 00110000 ⊕ 00101111 = 10100011 w5 = 10100011⊕ 11101001 = 01001010

The S-Box

The S-box is constructed as follows:

1. Initialize the S-box with the nibble values in ascending sequence row by row. The first row contains the hexadecimal values (0, 1, 2, 3); the second row con-tains (4, 5, 6, 7); and so on. Thus, the value of the nibble at row i, column j is 4i + j.

2. Treat each nibble as an element of the finite field (24) modulo x4 + x + 1. Each nibble a0 a1 a2 a3 represents a polynomial of degree 3.

3. Map each byte in the S-box to its multiplicative inverse in the finite field GF(24) modulo x4 + x + 1; the value 0 is mapped to itself.

4. Consider that each byte in the S-box consists of 4 bits labeled (b0, b1, b2, b3). Apply the following transformation to each bit of each byte in the S-box. The AES standard depicts this transformation in matrix form asD b′0

b′1b′2b′3

T = D1 0 1 11 1 0 11 1 1 00 1 1 1

T Db0

b1

b2

b3

T ⊕ D1001

T 5. The prime (′) indicates that the variable is to be updated by the value on

the right. Remember that addition and multiplication are being calculated modulo 2.

Table 5.7a shows the resulting S-box. This is a nonlinear, invertible matrix. The inverse S-box is shown in Table 5.7b.


S-AES Structure

We can now examine several aspects of interest concerning the structure of AES. First, note that the encryption and decryption algorithms begin and end with the add key function. Any other function, at the beginning or end, is easily reversible without knowledge of the key and so would add no security but just a processing overhead. Thus, there is a round 0 consisting of only the add key function.

The second point to note is that round 2 does not include the mix column function. The explanation for this in fact relates to a third observation, which is that although the decryption algorithm is the reverse of the encryption algo-rithm, as clearly seen in Figure 5.11, it does not follow the same sequence of functions. Thus,

Encryption: AK2∘ SR ∘ NS ∘ AK1


Decryption: AK0∘ INS ∘ ISR ∘ IMC ∘ AK1


From an implementation point of view, it would be desirable to have the decryption function follow the same function sequence as encryption. This allows the decryption algorithm to be implemented in the same way as the encryption algo-rithm, creating opportunities for efficiency.

Note that if we were able to interchange the second and third functions, the fourth and fifth functions, and the sixth and seventh functions in the decryption sequence, we would have the same structure as the encryption algorithm. Let’s see if this is possible. First, consider the interchange of INS and ISR. Given a state N con-sisting of the nibbles (N0, N1, N2, N3), the transformation INS(ISR(N)) proceeds as

aN0 N2

N1 N3b S aN0 N2

N3 N1b S aIS[N0] IS[N2]

IS[N3] IS[N1]b

Where IS refers to the inverse S-Box. Reversing the operations, the transfor-mation ISR(INS(N) proceeds as

aN0 N2

N1 N3b S aIS[N0] IS[N2]

IS[N1] IS[N3]b S aIS[N0] IS[N2]

IS[N3] IS[N1]b

which is the same result. Thus, INS(ISR(N)) = ISR(INS(N)).Now consider the operation of inverse mix column followed by add key

IMC(AK1(N)) where the round key K1 consists of the nibbles (k0,0, k1,0, k0,1, k1,1).

Then

a9 22 9

baak0,0 k0,1

k1,0 k1,1b ⊕ aN0 N2

N1 N3b b = a9 2

2 9bak0,0 ⊕ N0 k0,1 ⊕ N2

k1,0 ⊕ N1 k1,1 ⊕ N3b

= a9(k0,0 ⊕ N0) ⊕ 2(K1,0 ⊕ N1) 9(k0,1 ⊕ N2) ⊕ 2(K1,1 ⊕ N3)2(k0,0 ⊕ N0) ⊕ 9(K1,0 ⊕ N1) 2(k0,1 ⊕ N2) ⊕ 9(K1,1 ⊕ N3)

b

= a(9k0,0 ⊕ 2k1,0) ⊕ (9N0 ⊕ 2N1) (9k0,1 ⊕ 2k1,1) ⊕ (9N2 ⊕ 2N3)(2k0,0 ⊕ 9k1,0) ⊕ (2N0 ⊕ 9N1) (2k0,1 ⊕ 9k1,1) ⊕ (2N2 ⊕ 9N3)

b


= a(9k0,0 ⊕ 2k1,0) (9k0,1 ⊕ 2k1,1)(2k0,0 ⊕ 9k1,0) (2k0,1 ⊕ 9k1,1)

b ⊕ a(9N0 ⊕ 2N1) (9N2 ⊕ 2N3)(2N0 ⊕ 9N1) (2N2 ⊕ 9N3)

b

= a9 22 9

bak0,0 k0,1

k1,0 k1,1b ⊕ a9 2

2 9baN0 N2

N1 N3b

All of these steps make use of the properties of finite field arithmetic. The result is that IMC(AK1

(N)) = IMC(K1) ⊕ IMC(N). Now let us define the inverse round key for round 1 to be IMC(K1) and the inverse add key operation IAK1

to be the bitwise XOR of the inverse round key with the state vector. Then we have IMC(AK1

(N)) = IAK1(IMC(N)). As a result, we can write the following:

Encryption: AK2 ∘ SR ∘ NS ∘ AK1


Decryption: AK0 ∘ INS ∘ ISR ∘ IMC ∘ AK1


Decryption: AK0 ∘ ISR ∘ INS ∘ AIMC(K1) ∘ IMC ∘ ISR ∘ INS ∘ AK2

Both encryption and decryption now follow the same sequence. Note that this derivation would not work as effectively if round 2 of the encryption algorithm included the MC function. In that case, we would have

Encryption: AK2 ∘ MC ∘ SR ∘ NS ∘ AK1


Decryption: AK0 ∘ INS ∘ ISR ∘ IMC ∘ AK1

∘ INS ∘ ISR ∘ IMC ∘ AK2

There is now no way to interchange pairs of operations in the decryption algorithm so as to achieve the same structure as the encryption algorithm.

6.1 Multiple Encryption and Triple dEs

Double DESTriple DES with Two KeysTriple DES with Three Keys

6.2 Electronic Code Book

6.3 Cipher Block Chaining Mode

6.4 Cipher Feedback Mode

6.5 Output Feedback Mode

6.6 Counter Mode

6.7 XTs-AEs Mode for Block-Oriented storage devices

Storage Encryption RequirementsOperation on a Single BlockOperation on a Sector



Chapter

Block cipher operation

174174

6.1 / Multiple encryption and triple deS 175

Many savages at the present day regard their names as vital parts of themselves, and therefore take great pains to conceal their real names, lest these should give to evil-disposed persons a handle by which to injure their owners.


Learning Objectives


u Analyze the security of multiple encryption schemes.

u Explain the meet-in-the-middle attack.

u Compare and contrast ECB, CBC, CFB, OFB, and counter modes of operation.

u Present an overview of the XTS-AES mode of operation.

This chapter continues our discussion of symmetric ciphers. We begin with the topic of multiple encryption, looking in particular at the most widely used multiple-encryption scheme: triple DES.

The chapter next turns to the subject of block cipher modes of operation. We find that there are a number of different ways to apply a block cipher to plaintext, each with its own advantages and particular applications.

6.1 MuLtipLe encryptiOn and tripLe des

Given the potential vulnerability of DES to a brute-force attack, there has been considerable interest in finding an alternative. One approach is to design a com-pletely new algorithm, of which AES is a prime example. Another alternative, which would preserve the existing investment in software and equipment, is to use multiple encryption with DES and multiple keys. We begin by examining the sim-plest example of this second alternative. We then look at the widely accepted triple DES (3DES) approach.

Double DES

The simplest form of multiple encryption has two encryption stages and two keys (Figure 6.1a). Given a plaintext P and two encryption keys K1 and K2, ciphertext C is generated as

C = E(K2, E(K1, P))

Decryption requires that the keys be applied in reverse order:

P = D(K1, D(K2, C))

176 chapter 6 / Block cipher operation

For DES, this scheme apparently involves a key length of 56 * 2 = 112 bits, result-ing in a dramatic increase in cryptographic strength. But we need to examine the algorithm more closely.

Reduction to a Single Stage Suppose it were true for DES, for all 56-bit key values, that given any two keys K1 and K2, it would be possible to find a key K3 such that

E(K2, E(K1, P)) = E(K3, P) (6.1)

If this were the case, then double encryption, and indeed any number of stages of multiple encryption with DES, would be useless because the result would be equiv-alent to a single encryption with a single 56-bit key.

On the face of it, it does not appear that Equation (6.1) is likely to hold. Consider that encryption with DES is a mapping of 64-bit blocks to 64-bit blocks. In fact, the mapping can be viewed as a permutation. That is, if we consider all 264 possible input blocks, DES encryption with a specific key will map each block into a unique 64-bit block. Otherwise, if, say, two given input blocks mapped to the same output block, then decryption to recover the original plaintext would be impossible.

E E

K1

P

K2

CX

Encryption

D D

K2

C

K1

PX

Decryption

(a) Double encryption

E D E

K1

P

K2 K1

CA B

Encryption

D E D

K1

C

K2 K1

P

Decryption

(b) Triple encryption

B A

Figure 6.1 Multiple Encryption


With 264 possible inputs, how many different mappings are there that generate a permutation of the input blocks? The value is easily seen to be

(264)! = 10347380000000000000000 7 (101020)

On the other hand, DES defines one mapping for each different key, for a total number of mappings:

256 6 1017

Therefore, it is reasonable to assume that if DES is used twice with different keys, it will produce one of the many mappings that are not defined by a single application of DES. Although there was much supporting evidence for this assumption, it was not until 1992 that the assumption was proven [CAMP92].

Meet-in-the-Middle attack Thus, the use of double DES results in a mapping that is not equivalent to a single DES encryption. But there is a way to attack this scheme, one that does not depend on any particular property of DES but that will work against any block encryption cipher.

The algorithm, known as a meet-in-the-middle attack, was first described in [DIFF77]. It is based on the observation that, if we have

C = E(K2, E(K1, P))

then (see Figure 6.1a)

X = E(K1, P) = D(K2, C)

Given a known pair, (P, C), the attack proceeds as follows. First, encrypt P for all 256 possible values of K1. Store these results in a table and then sort the table by the values of X. Next, decrypt C using all 256 possible values of K2. As each decryption is produced, check the result against the table for a match. If a match occurs, then test the two resulting keys against a new known plaintext–ciphertext pair. If the two keys produce the correct ciphertext, accept them as the correct keys.

For any given plaintext P, there are 264 possible ciphertext values that could be produced by double DES. Double DES uses, in effect, a 112-bit key, so that there are 2112 possible keys. Therefore, on average, for a given plaintext P, the num-ber of different 112-bit keys that will produce a given ciphertext C is 2112/264 = 248. Thus, the foregoing procedure will produce about 248 false alarms on the first (P, C) pair. A similar argument indicates that with an additional 64 bits of known plaintext and ciphertext, the false alarm rate is reduced to 248 - 64 = 2-16. Put another way, if the meet-in-the-middle attack is performed on two blocks of known plaintext– ciphertext, the probability that the correct keys are determined is 1 - 2- 16. The result is that a known plaintext attack will succeed against double DES, which has a key size of 112 bits, with an effort on the order of 256, which is not much more than the 255 required for single DES.

Triple DES with Two Keys

An obvious counter to the meet-in-the-middle attack is to use three stages of encryp-tion with three different keys. This raises the cost of the meet-in-the-middle attack


to 2112, which is beyond what is practical now and far into the future. However, it has the drawback of requiring a key length of 56 * 3 = 168 bits, which may be somewhat unwieldy.

As an alternative, Tuchman proposed a triple encryption method that uses only two keys [TUCH79]. The function follows an encrypt-decrypt-encrypt (EDE) sequence (Figure 6.1b):

C = E(K1, D(K2, E(K1, P)))

P = D(K1, E(K2, D(K1, C)))

There is no cryptographic significance to the use of decryption for the second stage. Its only advantage is that it allows users of 3DES to decrypt data encrypted by users of the older single DES:

C = E(K1, D(K1, E(K1, P))) = E(K1, P)

P = D(K1, E(K1, D(K1, C))) = D(K1, C)

3DES with two keys is a relatively popular alternative to DES and has been adopted for use in the key management standards ANSI X9.17 and ISO 8732.1

Currently, there are no practical cryptanalytic attacks on 3DES. Coppersmith [COPP94] notes that the cost of a brute-force key search on 3DES is on the order of 2112 ≈ (5 * 1033) and estimates that the cost of differential cryptanalysis suffers an exponential growth, compared to single DES, exceeding 1052.

It is worth looking at several proposed attacks on 3DES that, although not practical, give a flavor for the types of attacks that have been considered and that could form the basis for more successful future attacks.

The first serious proposal came from Merkle and Hellman [MERK81]. Their plan involves finding plaintext values that produce a first intermediate value of A = 0 (Figure 6.1b) and then using the meet-in-the-middle attack to determine the two keys. The level of effort is 256, but the technique requires 256 chosen plain-text–ciphertext pairs, which is a number unlikely to be provided by the holder of the keys.

A known-plaintext attack is outlined in [VANO90]. This method is an im-provement over the chosen-plaintext approach but requires more effort. The attack is based on the observation that if we know A and C (Figure 6.1b), then the problem reduces to that of an attack on double DES. Of course, the attacker does not know A, even if P and C are known, as long as the two keys are unknown. However, the attacker can choose a potential value of A and then try to find a known (P, C) pair that produces A. The attack proceeds as follows.

1. Obtain n (P, C) pairs. This is the known plaintext. Place these in a table (Table 1) sorted on the values of P (Figure 6.2b).

1American National Standards Institute (ANSI): Financial Institution Key Management (Wholesale). From its title, X9.17 appears to be a somewhat obscure standard. Yet a number of techniques specified in this standard have been adopted for use in other standards and applications, as we shall see throughout this book.


2. Pick an arbitrary value a for A, and create a second table (Figure 6.2c) with en-tries defined in the following fashion. For each of the 256 possible keys K1 = i, calculate the plaintext value Pi that produces a:

Pi = D(i, a)

For each Pi that matches an entry in Table 1, create an entry in Table 2 consist-ing of the K1 value and the value of B that is produced for the (P, C) pair from Table 1, assuming that value of K1:

B = D(i, C)

At the end of this step, sort Table 2 on the values of B.

3. We now have a number of candidate values of K1 in Table 2 and are in a posi-tion to search for a value of K2. For each of the 256 possible keys K2 = j, calcu-late the second intermediate value for our chosen value of a:

Bj = D(j, a)

At each step, look up Bj in Table 2. If there is a match, then the corresponding key i from Table 2 plus this value of j are candidate values for the unknown keys (K1, K2). Why? Because we have found a pair of keys (i, j) that produce a known (P, C) pair (Figure 6.2a).

4. Test each candidate pair of keys (i, j) on a few other plaintext–ciphertext pairs. If a pair of keys produces the desired ciphertext, the task is complete. If no pair succeeds, repeat from step 1 with a new value of a.

E D E

i j i

Ci

a Bj

(a) Two-key triple encryption with candidate pair of keys

Pi

Pi Ci

(b) Table of n knownplaintext–ciphertext

pairs, sorted on P

Bj Key i

(c) Table of intermediatevalues and candidate

keys

Figure 6.2 Known-Plaintext Attack on Triple DES


For a given known (P, C), the probability of selecting the unique value of a that leads to success is 1/264. Thus, given n (P, C) pairs, the probability of success for a single selected value of a is n/264. A basic result from probability theory is that the expected number of draws required to draw one red ball out of a bin containing n red balls and N - n green balls is (N + 1)/(n + 1) if the balls are not replaced. So the expected number of values of a that must be tried is, for large n,

264 + 1n + 1

≈264

n

Thus, the expected running time of the attack is on the order of

12562 264

n= 2120 - log2n

Triple DES with Three Keys

Although the attacks just described appear impractical, anyone using two-key 3DES may feel some concern. Thus, many researchers now feel that three-key 3DES is the preferred alternative (e.g., [KALI96a]). Three-key 3DES has an effective key length of 168 bits and is defined as

C = E(K3, D(K2, E(K1, P)))

Backward compatibility with DES is provided by putting K3 = K2 or K1 = K2.A number of Internet-based applications have adopted three-key 3DES, in-

cluding PGP and S/MIME, both discussed in Chapter 19.

6.2 eLectrOnic cOde bOOk

A block cipher takes a fixed-length block of text of length b bits and a key as input and produces a b-bit block of ciphertext. If the amount of plaintext to be encrypted is greater than b bits, then the block cipher can still be used by breaking the plain-text up into b-bit blocks. When multiple blocks of plaintext are encrypted using the same key, a number of security issues arise. To apply a block cipher in a variety of applications, five modes of operation have been defined by NIST (SP 800-38A). In essence, a mode of operation is a technique for enhancing the effect of a cryp-tographic algorithm or adapting the algorithm for an application, such as applying a block cipher to a sequence of data blocks or a data stream. The five modes are intended to cover a wide variety of applications of encryption for which a block cipher could be used. These modes are intended for use with any symmetric block cipher, including triple DES and AES. The modes are summarized in Table 6.1 and described in this and the following sections.

The simplest mode is the electronic codebook (ECB) mode, in which plaintext is handled one block at a time and each block of plaintext is encrypted using the same key (Figure 6.3). The term codebook is used because, for a given key, there is a unique ciphertext for every b-bit block of plaintext. Therefore, we can imagine a gigantic codebook in which there is an entry for every possible b-bit plaintext pat-tern showing its corresponding ciphertext.

6.2 / electronic code Book 181

For a message longer than b bits, the procedure is simply to break the message into b-bit blocks, padding the last block if necessary. Decryption is performed one block at a time, always using the same key. In Figure 6.3, the plaintext (padded as necessary) consists of a sequence of b-bit blocks, P1, P2, c , PN; the correspond-ing sequence of ciphertext blocks is C1, C2, c , CN. We can define ECB mode as follows.

ECB C j = E(K, Pj) j = 1, c , N Pj = D(K, Cj) j = 1, c , N

The ECB method is ideal for a short amount of data, such as an encryption key. Thus, if you want to transmit a DES or AES key securely, ECB is the appropri-ate mode to use.

The most significant characteristic of ECB is that if the same b-bit block of plaintext appears more than once in the message, it always produces the same ciphertext.

For lengthy messages, the ECB mode may not be secure. If the message is highly structured, it may be possible for a cryptanalyst to exploit these regulari-ties. For example, if it is known that the message always starts out with certain predefined fields, then the cryptanalyst may have a number of known plaintext– ciphertext pairs to work with. If the message has repetitive elements with a period of repetition a multiple of b bits, then these elements can be identified by the analyst. This may help in the analysis or may provide an opportunity for substituting or rearranging blocks.

Table 6.1 Block Cipher Modes of Operation

Mode description Typical Application

Electronic Codebook (ECB) Each block of plaintext bits is encoded independently using the same key.

• Secure transmission of single values (e.g., an encryption key)

Cipher Block Chaining (CBC) The input to the encryption algo-rithm is the XOR of the next block of plaintext and the preceding block of ciphertext.

• General-purpose block-oriented transmission

• Authentication

Cipher Feedback (CFB) Input is processed s bits at a time. Preceding ciphertext is used as input to the encryption algorithm to produce pseudorandom output, which is XORed with plaintext to produce next unit of ciphertext.

• General-purpose stream-oriented transmission

• Authentication

Output Feedback (OFB) Similar to CFB, except that the input to the encryption algorithm is the preceding encryption output, and full blocks are used.

• Stream-oriented transmission over noisy channel (e.g., satellite communication)

Counter (CTR) Each block of plaintext is XORed with an encrypted counter. The counter is incremented for each subsequent block.

• General-purpose block- oriented transmission

• Useful for high-speed requirements


We now turn to more complex modes of operation. [KNUD00] lists the fol-lowing criteria and properties for evaluating and constructing block cipher modes of operation that are superior to ECB:

• Overhead: The additional operations for the encryption and decryption operation when compared to encrypting and decrypting in the ECB mode.

• Error recovery: The property that an error in the ith ciphertext block is inher-ited by only a few plaintext blocks after which the mode resynchronizes.

• Error propagation: The property that an error in the ith ciphertext block is inherited by the ith and all subsequent plaintext blocks. What is meant here is a bit error that occurs in the transmission of a ciphertext block, not a compu-tational error in the encryption of a plaintext block.

• diffusion: How the plaintext statistics are reflected in the ciphertext. Low entropy plaintext blocks should not be reflected in the ciphertext blocks. Roughly, low entropy equates to predictability or lack of randomness (see Appendix F).

• security: Whether or not the ciphertext blocks leak information about the plaintext blocks.

C1

P1

Encrypt

K

P2

C2

Encrypt

K

P N

CN

Encrypt

K

(a) Encryption

P1

C1

Decrypt

K

C2

P2

Decrypt

K

CN

PN

Decrypt

K

(b) Decryption

Figure 6.3 Electronic Codebook (ECB) Mode

6.3 / cipher Block chaining Mode 183

6.3 cipher bLOck chaining MOde

To overcome the security deficiencies of ECB, we would like a technique in which the same plaintext block, if repeated, produces different ciphertext blocks. A simple way to satisfy this requirement is the cipher block chaining (CBC) mode (Figure 6.4). In this scheme, the input to the encryption algorithm is the XOR of the current plaintext block and the preceding ciphertext block; the same key is used for each block. In effect, we have chained together the processing of the sequence of plaintext blocks. The input to the encryption function for each plaintext block bears no fixed relationship to the plaintext block. Therefore, repeating patterns of b bits are not exposed. As with the ECB mode, the CBC mode requires that the last block be padded to a full b bits if it is a partial block.

For decryption, each cipher block is passed through the decryption algorithm. The result is XORed with the preceding ciphertext block to produce the plaintext block. To see that this works, we can write

Cj = E(K, [Cj- 1 ⊕ Pj])

C1

P1

Encrypt

IV

K

P2

C2

Encrypt

K

PN

CN

CN–1

Encrypt

K

(a) Encryption

P1

C1

Decrypt

IV

K

C2

P2

Decrypt

K

CN

PN

CN–1

Decrypt

K

(b) Decryption

Figure 6.4 Cipher Block Chaining (CBC) Mode


2NIST SP-800-90 (Recommendation for Random Number Generation Using Deterministic Random Bit Generators) defines nonce as follows: A time-varying value that has at most a negligible chance of repeat-ing, for example, a random value that is generated anew for each use, a timestamp, a sequence number, or some combination of these.

Then

D(K, Cj) = D(K, E(K, [Cj- 1 ⊕ Pj]))

D(K, Cj) = Cj- 1 ⊕ Pj

Cj- 1 ⊕ D(K, Cj) = Cj- 1 ⊕ Cj- 1 ⊕ Pj = Pj

To produce the first block of ciphertext, an initialization vector (IV) is XORed with the first block of plaintext. On decryption, the IV is XORed with the output of the decryption algorithm to recover the first block of plaintext. The IV is a data block that is the same size as the cipher block. We can define CBC mode as

CBC C1 = E(K, [P1 ⊕ IV])

Cj = E(K, [Pj ⊕ Cj- 1]) j = 2, c , N

P1 = D(K, C1) ⊕ IV

Pj = D(K, Cj) ⊕ Cj- 1 j = 2, c , N

The IV must be known to both the sender and receiver but be unpredictable by a third party. In particular, for any given plaintext, it must not be possible to predict the IV that will be associated to the plaintext in advance of the generation of the IV. For maximum security, the IV should be protected against unauthorized changes. This could be done by sending the IV using ECB encryption. One reason for protecting the IV is as follows: If an opponent is able to fool the receiver into using a different value for IV, then the opponent is able to invert selected bits in the first block of plaintext. To see this, consider

C1 = E(K, [IV ⊕ P1])

P1 = IV ⊕ D(K, C1)

Now use the notation that X[i] denotes the ith bit of the b-bit quantity X. Then

P1[i] = IV[i] ⊕ D(K, C1)[i]

Then, using the properties of XOR, we can state

P1[i]′ = IV[i]′ ⊕ D(K, C1)[i]

where the prime notation denotes bit complementation. This means that if an oppo-nent can predictably change bits in IV, the corresponding bits of the received value of P1 can be changed.

For other possible attacks based on prior knowledge of IV, see [VOYD83].So long as it is unpredictable, the specific choice of IV is unimportant.

SP800-38A recommends two possible methods: The first method is to apply the encryption function, under the same key that is used for the encryption of the plain-text, to a nonce.2 The nonce must be a data block that is unique to each execution of the encryption operation. For example, the nonce may be a counter, a timestamp, or

6.4 / cipher FeedBack Mode 185

a message number. The second method is to generate a random data block using a random number generator.

In conclusion, because of the chaining mechanism of CBC, it is an appropriate mode for encrypting messages of length greater than b bits.

In addition to its use to achieve confidentiality, the CBC mode can be used for authentication. This use is described in Chapter 12.

6.4 cipher Feedback MOde

For AES, DES, or any block cipher, encryption is performed on a block of b bits. In the case of DES, b = 64 and in the case of AES, b = 128. However, it is possible to convert a block cipher into a stream cipher, using one of the three modes to be discussed in this and the next two sections: cipher feedback (CFB) mode, output feedback (OFB) mode, and counter (CTR) mode. A stream cipher eliminates the need to pad a message to be an integral number of blocks. It also can operate in real time. Thus, if a character stream is being transmitted, each character can be encrypted and transmitted immediately using a character-oriented stream cipher.

One desirable property of a stream cipher is that the ciphertext be of the same length as the plaintext. Thus, if 8-bit characters are being transmitted, each charac-ter should be encrypted to produce a ciphertext output of 8 bits. If more than 8 bits are produced, transmission capacity is wasted.

Figure 6.5 depicts the CFB scheme. In the figure, it is assumed that the unit of transmission is s bits; a common value is s = 8. As with CBC, the units of plaintext are chained together, so that the ciphertext of any plaintext unit is a function of all the preceding plaintext. In this case, rather than blocks of b bits, the plaintext is divided into segments of s bits.

First, consider encryption. The input to the encryption function is a b-bit shift register that is initially set to some initialization vector (IV). The leftmost (most significant) s bits of the output of the encryption function are XORed with the first segment of plaintext P1 to produce the first unit of ciphertext C1, which is then transmitted. In addition, the contents of the shift register are shifted left by s bits, and C1 is placed in the rightmost (least significant) s bits of the shift register. This process continues until all plaintext units have been encrypted.

For decryption, the same scheme is used, except that the received ciphertext unit is XORed with the output of the encryption function to produce the plaintext unit. Note that it is the encryption function that is used, not the decryption func-tion. This is easily explained. Let MSBs(X) be defined as the most significant s bits of X. Then

C1 = P1 ⊕ MSBs[E(K, IV)]

Therefore, by rearranging terms:

P1 = C1 ⊕ MSBs[E(K, IV)]

The same reasoning holds for subsequent steps in the process.


We can define CFB mode as follows.

CFB

I1 = IV

Ij = LSBb - s(Ij- 1) }Cj- 1 j = 2, c , N

Oj = E(K, Ij) j = 1, c , N

Cj = Pj ⊕ MSBs(Oj) j = 1, c , N

I1 = IV

Ij = LSBb - s(Ij- 1) }Cj- 1 j = 2, c , N

Oj = E(K, Ij) j = 1, c , N

Pj = Cj ⊕ MSBs(Oj) j = 1, c , N

Although CFB can be viewed as a stream cipher, it does not conform to the typical construction of a stream cipher. In a typical stream cipher, the cipher takes

C1

IV

P1

Encrypt

Select s bits

Discardb – s bits

K

(a) Encryption

CN–1

(b) Decryption

s bits

s bits s bits

C2

P2

Encrypt

Selects bits

Discardb – s bits

K

s bits

s bitsb – s bitsShift register

s bits

CN

PN

Encrypt

Selects bits

Discardb – s bits

K

s bits


P1

IV

C1

Encrypt

Select s bits

Discardb – s bits

K

CN–1

s bits

C2

s bits

CN

s bits

s bits s bits

P2

Encrypt

Selects bits

Discardb – s bits

K



s bits

PN

Encrypt

Selects bits

Discardb – s bits

K

Figure 6.5 s-bit Cipher Feedback (CFB) Mode

6.5 / output FeedBack Mode 187

as input some initial value and a key and generates a stream of bits, which is then XORed with the plaintext bits (see Figure 3.1). In the case of CFB, the stream of bits that is XORed with the plaintext also depends on the plaintext.

In CFB encryption, like CBC encryption, the input block to each forward cipher function (except the first) depends on the result of the previous forward cipher function; therefore, multiple forward cipher operations cannot be performed in parallel. In CFB decryption, the required forward cipher operations can be per-formed in parallel if the input blocks are first constructed (in series) from the IV and the ciphertext.

6.5 Output Feedback MOde

The output feedback (OFB) mode is similar in structure to that of CFB. For OFB, the output of the encryption function is fed back to become the input for encrypting the next block of plaintext (Figure 6.6). In CFB, the output of the XOR unit is fed back to become input for encrypting the next block. The other difference is that the OFB mode operates on full blocks of plaintext and ciphertext, whereas CFB oper-ates on an s-bit subset. OFB encryption can be expressed as

Cj = Pj ⊕ E(K, Oj- 1)

where

Oj- 1 = E(K, Oj- 2)

Some thought should convince you that we can rewrite the encryption expres-sion as:

Cj = Pj ⊕ E(K, [Cj- 1 ⊕ Pj- 1])

By rearranging terms, we can demonstrate that decryption works.

Pj = Cj ⊕ E(K, [Cj- 1 ⊕ Pj- 1])

We can define OFB mode as follows.

OFB

I1 = Nonce

Ij = Oj- 1 j = 2, c , N

Oj = E(K, Ij) j = 1, c , N

Cj = Pj ⊕ Oj j = 1, c , N - 1

C*N = P*

N ⊕ MSBu(ON)

I1 = Nonce

Ij = Oj- 1 j = 2, c , N

Oj = E(K, Ij) j = 1, c , N

Pj = Cj ⊕ Oj j = 1, c , N - 1

P*N = C*

N ⊕ MSBu(ON)

Let the size of a block be b. If the last block of plaintext contains u bits (indi-cated by *), with u 6 b, the most significant u bits of the last output block ON are used for the XOR operation; the remaining b - u bits of the last output block are discarded.

As with CBC and CFB, the OFB mode requires an initialization vector. In the case of OFB, the IV must be a nonce; that is, the IV must be unique to each execution of the encryption operation. The reason for this is that the sequence of


encryption output blocks, Oi, depends only on the key and the IV and does not de-pend on the plaintext. Therefore, for a given key and IV, the stream of output bits used to XOR with the stream of plaintext bits is fixed. If two different messages had an identical block of plaintext in the identical position, then an attacker would be able to determine that portion of the Oi stream.

One advantage of the OFB method is that bit errors in transmission do not propagate. For example, if a bit error occurs in C1, only the recovered value of P1 is affected; subsequent plaintext units are not corrupted. With CFB, C1 also serves as input to the shift register and therefore causes additional corruption downstream.

The disadvantage of OFB is that it is more vulnerable to a message stream modification attack than is CFB. Consider that complementing a bit in the cipher-text complements the corresponding bit in the recovered plaintext. Thus, controlled

(a) Encryption

P1

C1

Nonce

Encrypt

K

P2 PN

C2

Encrypt

K

CN

Encrypt

K

(b) Decryption

C1

P1

Nonce

Encrypt

K

C2 CN

P2

Encrypt

K

PN

Encrypt

K

Figure 6.6 Output Feedback (OFB) Mode

6.6 / counter Mode 189

changes to the recovered plaintext can be made. This may make it possible for an opponent, by making the necessary changes to the checksum portion of the message as well as to the data portion, to alter the ciphertext in such a way that it is not de-tected by an error-correcting code. For a further discussion, see [VOYD83].

OFB has the structure of a typical stream cipher, because the cipher gener-ates a stream of bits as a function of an initial value and a key, and that stream of bits is XORed with the plaintext bits (see Figure 3.1). The generated stream that is XORed with the plaintext is itself independent of the plaintext; this is highlighted by dashed boxes in Figure 6.6. One distinction from the stream ciphers we discuss in Chapter 7 is that OFB encrypts plaintext a full block at a time, where typically a block is 64 or 128 bits. Many stream ciphers encrypt one byte at a time.

6.6 cOunter MOde

Although interest in the counter (CTR) mode has increased recently with applica-tions to ATM (asynchronous transfer mode) network security and IP sec (IP secu-rity), this mode was proposed early on (e.g., [DIFF79]).

Figure 6.7 depicts the CTR mode. A counter equal to the plaintext block size is used. The only requirement stated in SP 800-38A is that the counter value must be different for each plaintext block that is encrypted. Typically, the counter is initialized to some value and then incremented by 1 for each subsequent block (modulo 2b, where b is the block size). For encryption, the counter is encrypted and then XORed with the plaintext block to produce the ciphertext block; there is no chaining. For decryption, the same sequence of counter values is used, with each en-crypted counter XORed with a ciphertext block to recover the corresponding plain-text block. Thus, the initial counter value must be made available for decryption. Given a sequence of counters T1, T2, c , TN, we can define CTR mode as follows.

CTR Cj = Pj ⊕ E(K, Tj) j = 1, c , N - 1

C*N = P*

N ⊕ MSBu[E(K, TN)]

Pj = Cj ⊕ E(K, Tj) j = 1, c , N - 1

P*N = C*

N ⊕ MSBu[E(K, TN)]

For the last plaintext block, which may be a partial block of u bits, the most significant u bits of the last output block are used for the XOR operation; the re-maining b - u bits are discarded. Unlike the ECB, CBC, and CFB modes, we do not need to use padding because of the structure of the CTR mode.

As with the OFB mode, the initial counter value must be a nonce; that is, T1 must be different for all of the messages encrypted using the same key. Further, all Ti values across all messages must be unique. If, contrary to this requirement, a counter value is used multiple times, then the confidentiality of all of the plaintext blocks corresponding to that counter value may be compromised. In particular, if any plaintext block that is encrypted using a given counter value is known, then the output of the encryption function can be determined easily from the associated ciphertext block. This output allows any other plaintext blocks that are encrypted using the same counter value to be easily recovered from their associated cipher-text blocks.


One way to ensure the uniqueness of counter values is to continue to incre-ment the counter value by 1 across messages. That is, the first counter value of the each message is one more than the last counter value of the preceding message.

[LIPM00] lists the following advantages of CTR mode.

• Hardware efficiency: Unlike the three chaining modes, encryption (or decryp-tion) in CTR mode can be done in parallel on multiple blocks of plaintext or ciphertext. For the chaining modes, the algorithm must complete the computa-tion on one block before beginning on the next block. This limits the maximum throughput of the algorithm to the reciprocal of the time for one execution of block encryption or decryption. In CTR mode, the throughput is only limited by the amount of parallelism that is achieved.

(a) Encryption

P1

C1

Counter 1

Encrypt

K

Counter 2 Counter N

P2 PN

C2

Encrypt

K

CN

Encrypt

K

(b) Decryption

C1

P1

Counter 1

Encrypt

K

Counter 2 Counter N

C2 CN

P2

Encrypt

K

PN

Encrypt

K

Figure 6.7 Counter (CTR) Mode

6.7 / XtS-aeS Mode For Block-oriented Storage deviceS 191

• software efficiency: Similarly, because of the opportunities for parallel execu-tion in CTR mode, processors that support parallel features, such as aggres-sive pipelining, multiple instruction dispatch per clock cycle, a large number of registers, and SIMD instructions, can be effectively utilized.

• Preprocessing: The execution of the underlying encryption algorithm does not depend on input of the plaintext or ciphertext. Therefore, if sufficient memory is available and security is maintained, preprocessing can be used to prepare the output of the encryption boxes that feed into the XOR functions, as in Figure 6.7. When the plaintext or ciphertext input is presented, then the only computation is a series of XORs. Such a strategy greatly enhances throughput.

• Random access: The ith block of plaintext or ciphertext can be processed in random-access fashion. With the chaining modes, block Ci cannot be com-puted until the i - 1 prior block are computed. There may be applications in which a ciphertext is stored and it is desired to decrypt just one block; for such applications, the random access feature is attractive.

• Provable security: It can be shown that CTR is at least as secure as the other modes discussed in this section.

• simplicity: Unlike ECB and CBC modes, CTR mode requires only the im-plementation of the encryption algorithm and not the decryption algorithm. This matters most when the decryption algorithm differs substantially from the encryption algorithm, as it does for AES. In addition, the decryption key scheduling need not be implemented.

Note that, with the exception of ECB, all of the NIST-approved block cipher modes of operation involve feedback. This is clearly seen in Figure 6.8. To highlight the feedback mechanism, it is useful to think of the encryption function as taking input from a input register whose length equals the encryption block length and with output stored in an output register. The input register is updated one block at a time by the feedback mechanism. After each update, the encryp-tion algorithm is executed, producing a result in the output register. Meanwhile, a block of plaintext is accessed. Note that both OFB and CTR produce output that is independent of both the plaintext and the ciphertext. Thus, they are natu-ral candidates for stream ciphers that encrypt plaintext by XOR one full block at a time.

6.7 Xts-aes MOde FOr bLOck-Oriented stOrage devices

In 2010, NIST approved an additional block cipher mode of operation, XTS-AES. This mode is also an IEEE standard, IEEE Std 1619-2007, which was developed by the IEEE Security in Storage Working Group (P1619). The standard describes a method of encryption for data stored in sector-based devices where the threat model includes possible access to stored data by the adversary. The standard has received widespread industry support.


Tweakable Block Ciphers

The XTS-AES mode is based on the concept of a tweakable block cipher, intro-duced in [LISK02]. The form of this concept used in XTS-AES was first described in [ROGA04].

Before examining XTS-AES, let us consider the general structure of a tweak-able block cipher. A tweakable block cipher is one that has three inputs: a plaintext P, a symmetric key K, and a tweak T; and produces a ciphertext output C. We can write this as C = E(K, T, P). The tweak need not be kept secret. Whereas the pur-pose of the key is to provide security, the purpose of the tweak is to provide vari-ability. That is, the use of different tweaks with the same plaintext and same key

Plaintext block

Plaintext block

Encrypt

Input register

Output register

Ciphertext Ciphertext

(a) Cipher block chaining (CBC) mode

Key

Encrypt

Input register

Output register

Key

(b) Cipher feedback (CFB) mode

Plaintext block

Ciphertext

Key

Encrypt

Input register

Output register

(c) Output feedback (OFB) mode

Plaintext block

Ciphertext

Key

Encrypt

Input register

Output register

Counter

(d) Counter (CTR) mode

Figure 6.8 Feedback Characteristic of Modes of Operation


produces different outputs. The basic structure of several tweakable clock ciphers that have been implemented is shown in Figure 6.9. Encryption can be expressed as:

C = H(T) ⊕ E(K, H(T) ⊕ P)

where H is a hash function. For decryption, the same structure is used with the plaintext as input and decryption as the function instead of encryption. To see that this works, we can write

H(T ) ⊕ C = E(K, H(T ) ⊕ P)

D[K, H(T ) ⊕ C] = H(T ) ⊕ P

H(T ) ⊕ D (K, H(T ) ⊕ C) = P

It is now easy to construct a block cipher mode of operation by using a differ-ent tweak value on each block. In essence, the ECB mode is used but for each block the tweak is changed. This overcomes the principal security weakness of ECB, which is that two encryptions of the same block yield the same ciphertext.

Storage Encryption Requirements

The requirements for encrypting stored data, also referred to as “data at rest” dif-fer somewhat from those for transmitted data. The P1619 standard was designed to have the following characteristics:

1. The ciphertext is freely available for an attacker. Among the circumstances that lead to this situation:

a. A group of users has authorized access to a database. Some of the records in the database are encrypted so that only specific users can successfully read/write them. Other users can retrieve an encrypted record but are un-able to read it without the key.

b. An unauthorized user manages to gain access to encrypted records. c. A data disk or laptop is stolen, giving the adversary access to the encrypted

data.

K

Hashfunction

Tj

HTj

Pj

Cj

Encrypt

(a) Encryption

K

Hashfunction

Tj Cj

Pj

Decrypt

(b) Decryption

Figure 6.9 Tweakable Block Cipher


2. The data layout is not changed on the storage medium and in transit. The en-crypted data must be the same size as the plaintext data.

3. Data are accessed in fixed sized blocks, independently from each other. That is, an authorized user may access one or more blocks in any order.

4. Encryption is performed in 16-byte blocks, independently from other blocks (except the last two plaintext blocks of a sector, if its size is not a multiple of 16 bytes).

5. There are no other metadata used, except the location of the data blocks within the whole data set.

6. The same plaintext is encrypted to different ciphertexts at different locations, but always to the same ciphertext when written to the same location again.

7. A standard conformant device can be constructed for decryption of data en-crypted by another standard conformant device.

The P1619 group considered some of the existing modes of operation for use with stored data. For CTR mode, an adversary with write access to the encrypted media can flip any bit of the plaintext simply by flipping the corresponding ciphertext bit.

Next, consider requirement 6 and the use of CBC. To enforce the requirement that the same plaintext encrypt to different ciphertext in different locations, the IV could be derived from the sector number. Each sector contains multiple blocks. An adversary with read/write access to the encrypted disk can copy a ciphertext sec-tor from one position to another, and an application reading the sector off the new location will still get the same plaintext sector (except perhaps the first 128 bits). For example, this means that an adversary that is allowed to read a sector from the second position but not the first can find the content of the sector in the first posi-tion by manipulating the ciphertext. Another weakness is that an adversary can flip any bit of the plaintext by flipping the corresponding ciphertext bit of the previous block, with the side-effect of “randomizing” the previous block.

Operation on a Single Block

Figure 6.10 shows the encryption and decryption of a single block. The operation in-volves two instances of the AES algorithm with two keys. The following parameters are associated with the algorithm.

Key The 256 or 512 bit XTS-AES key; this is parsed as a concatenation of two fields of equal size called Key1 and Key2, such that Key = Key1 }Key2.

Pj The jth block of plaintext. All blocks except possibly the final block have a length of 128 bits. A plaintext data unit, typically a disk sector, consists of a sequence of plaintext blocks P1, P2, c , Pm.

Cj The jth block of ciphertext. All blocks except possibly the final block have a length of 128 bits.

j The sequential number of the 128-bit block inside the data unit.

i The value of the 128-bit tweak. Each data unit (sector) is assigned a tweak value that is a nonnegative integer. The tweak values are assigned consecutively, starting from an arbitrary nonnegative integer.


a A primitive element of GF(2128) that corresponds to polynomial x (i.e., 0000c0102).

aja multiplied by itself j times, in GF(2128).

⊕ Bitwise XOR.

⊗ Modular multiplication of two polynomials with binary coefficients modulo x128 + x7 + x2 + x + 1. Thus, this is multiplication in GF(2128).

In essence, the parameter j functions much like the counter in CTR mode. It assures that if the same plaintext block appears at two different positions within a data unit, it will encrypt to two different ciphertext blocks. The parameter i func-tions much like a nonce at the data unit level. It assures that, if the same plaintext

Key2

Key1

AESEncrypt

i

T

CC

PP

Pj

Cj

AESEncrypt

(a) Encryption

(b) Decryption

j

Key2

Key1

AESEncrypt

i

T

CC

PP

Cj

Pj

AESDecrypt

j

Figure 6.10 XTS-AES Operation on Single Block


block appears at the same position in two different data units, it will encrypt to two different ciphertext blocks. More generally, it assures that the same plaintext data unit will encrypt to two different ciphertext data units for two different data unit positions.

The encryption and decryption of a single block can be described as

XTS-AES block operation

T = E(K2, i) ⊗ a j

PP = P ⊕ T

CC = E(K1, PP) C = CC ⊕ T

T = E(K2, i) ⊗ a j

CC = C ⊕ T

PP = D(K1, CC) P = PP ⊕ T

To see that decryption recovers the plaintext, let us expand the last line of both en-cryption and decryption. For encryption, we have

C = CC ⊕ T = E(K1, PP) ⊕ T = E(K1, P ⊕ T) ⊕ T

and for decryption, we have

P = PP ⊕ T = D(K1, CC) ⊕ T = D(K1, C ⊕ T) ⊕ T

Now, we substitute for C:

P = D(K1, C ⊕ T ) ⊕ T

= D(K1, [E(K1, P ⊕ T ) ⊕ T ] ⊕ T ) ⊕ T

= D(K1, E(K1, P ⊕ T )) ⊕ T

= (P ⊕ T ) ⊕ T = P

Operation on a Sector

The plaintext of a sector or data unit is organized into blocks of 128 bits. Blocks are labeled P0, P1, c, Pm. The last block my be null or may contain from 1 to 127 bits. In other words, the input to the XTS-AES algorithm consists of m 128-bit blocks and possibly a final partial block.

For encryption and decryption, each block is treated independently and encrypted/decrypted as shown in Figure 6.10. The only exception occurs when the last block has less than 128 bits. In that case, the last two blocks are en-crypted/decrypted using a ciphertext-stealing technique instead of padding. Figure 6.11 shows the scheme. Pm - 1 is the last full plaintext block, and Pm is the final plaintext block, which contains s bits with 1 … s … 127. Cm - 1 is the last full ciphertext block, and Cm is the final ciphertext block, which contains s bits. This technique is commonly called ciphertext stealing because the processing of the last block “steals” a temporary ciphertext of the penultimate block to complete the cipher block.

Let us label the block encryption and decryption algorithms of Figure 6.10 as

Block encryption: XTS-AES-blockEnc(K, Pj, i, j)Block decryption: XTS-AES-blockDec(K, Cj, i, j)


Then, XTS-AES mode is defined as follows:

XTS-AES mode with null final block

Cj = XTS@AES@blockEnc(K, Pj, i, j) j = 0, c, m - 1

Pj = XTS@AES@blockEnc(K, Cj, i, j) j = 0, c, m - 1

XTS-AES mode with final block containing s bits

Cj = XTS@AES@blockEnc(K, Pj, i, j) j = 0, c, m - 2XX = XTS@AES@blockEnc(K, Pm - 1, i, m - 1)CP = LSB128 - s(XX)YY = Pm }CP

Cm - 1 = XTS@AES@blockEnc(K, YY, i, m)Cm = MSBs(XX)

Pj = XTS@AES@blockDec(K, Cj, i, j) j = 0, c, m - 2YY = XTS@AES@blockDec(K, Cm - 1, i, m - 1)CP = LSB128 - s(YY)XX = Cm }CP

Pm - 1 = XTS@AES@blockDec(K, XX, i, m)Pm = MSBs(YY)

C0

P0

XTS-AESblock

encryption

Key

i, 0

C1

P1

XTS-AESblock

encryption

Key

i, 1

CP

XX

XX

YY

YY

Cm

CPPmPm–1

XTS-AESblock

encryption

Key

i, m–1

Cm–1

Cm–1

XTS-AESblock

encryption

Key

i, m

Cm

(a) Encryption

(b) Decryption

P0

C0

XTS-AESblock

decryption

Key

i, 0

P1

C1

XTS-AESblock

decryption

Key

i, 1

CPPm

CPCmCm–1

XTS-AESblock

decryption

Key

i, m

Pm–1

Pm–1

XTS-AESblock

decryption

Key

i, m–1

Pm

Figure 6.11 XTS-AES Mode


As can be seen, XTS-AES mode, like CTR mode, is suitable for parallel oper-ation. Because there is no chaining, multiple blocks can be encrypted or decrypted simultaneously. Unlike CTR mode, XTS-AES mode includes a nonce (the param-eter i) as well as a counter (parameter j).

6.8 recOMMended reading

[BALL12] provides a clear description of XTS-AES and examines its security properties.

BALL12 Ball, M., et al. “The XTS-AES Disk Encryption Algorithm and the Security of Ciphertext Stealing.” Cryptologia, January 2012.

6.9 key terMs, review QuestiOns, and prObLeMs

Key Terms

block cipher modes of operation

cipher block chaining mode (CBC)

cipher feedback mode (CFB)

ciphertext stealingcounter mode (CTR)electronic codebook mode

(ECB)meet-in-the-middle attack nonce

output feedback mode (OFB)

Triple DES (3DES)tweakable block cipherXTS-AES mode

Review Questions 6.1 What is triple encryption? 6.2 What is a meet-in-the-middle attack? 6.3 How many keys are used in triple encryption? 6.4 Why is the middle portion of 3DES a decryption rather than an encryption? 6.5 Why do some block cipher modes of operation only use encryption while others use

both encryption and decryption?

Problems 6.1 You want to build a hardware device to do block encryption in the cipher block chain-

ing (CBC) mode using an algorithm stronger than DES. 3DES is a good candidate. Figure 6.12 shows two possibilities, both of which follow from the definition of CBC. Which of the two would you choose:

a. For security? b. For performance? 6.2 Can you suggest a security improvement to either option in Figure 6.12, using only

three DES chips and some number of XOR functions? Assume you are still limited to two keys.

6.9 / key terMS, review QueStionS, and proBleMS 199

6.3 The Merkle-Hellman attack on 3DES begins by assuming a value of A = 0 (Figure 6.1b). Then, for each of the 256 possible values of K1, the plaintext P that produces A = 0 is determined. Describe the rest of the algorithm.

6.4 With the ECB mode, if there is an error in a block of the transmitted ciphertext, only the corresponding plaintext block is affected. However, in the CBC mode, this error propagates. For example, an error in the transmitted C1 (Figure 6.4) obviously cor-rupts P1 and P2.

a. Are any blocks beyond P2 affected? b. Suppose that there is a bit error in the source version of P1. Through how many

ciphertext blocks is this error propagated? What is the effect at the receiver? 6.5 Is it possible to perform encryption operations in parallel on multiple blocks of plain-

text in CBC mode? How about decryption? 6.6 CBC-Pad is a block cipher mode of operation used in the RC5 block cipher, but it

could be used in any block cipher. CBC-Pad handles plaintext of any length. The ciphertext is longer then the plaintext by at most the size of a single block. Padding is used to assure that the plaintext input is a multiple of the block length. It is assumed that the original plaintext is an integer number of bytes. This plaintext is padded at the end by from 1 to bb bytes, where bb equals the block size in bytes. The pad bytes are all the same and set to a byte that represents the number of bytes of padding. For example, if there are 8 bytes of padding, each byte has the bit pattern 00001000. Why not allow zero bytes of padding? That is, if the original plaintext is an integer multiple of the block size, why not refrain from padding?

Pn

K1 E

An�1

An

K2 D

K3 E

Bn�1

BnCn�1

Cn

(b) Three-loop CBC

Pn

K1, K2 EDE

Cn�1

Cn

(a) One-loop CBC

Figure 6.12 Use of Triple DES in CBC Mode


6.7 For the ECB, CBC, and CFB modes, the plaintext must be a sequence of one or more complete data blocks (or, for CFB mode, data segments). In other words, for these three modes, the total number of bits in the plaintext must be a positive multiple of the block (or segment) size. One common method of padding, if needed, consists of a 1 bit followed by as few zero bits, possibly none, as are necessary to complete the final block. It is considered good practice for the sender to pad every message, including messages in which the final message block is already complete. What is the motiva-tion for including a padding block when padding is not needed?

6.8 If a bit error occurs in the transmission of a ciphertext character in 8-bit CFB mode, how far does the error propagate?

6.9 In discussing OFB, it was mentioned that if it was known that two different messages had an identical block of plaintext in the identical position, it is possible to recover the corresponding Oi block. Show the calculation.

6.10 In discussing the CTR mode, it was mentioned that if any plaintext block that is en-crypted using a given counter value is known, then the output of the encryption function can be determined easily from the associated ciphertext block. Show the calculation.

6.11 Padding may not always be appropriate. For example, one might wish to store the en-crypted data in the same memory buffer that originally contained the plaintext. In that case, the ciphertext must be the same length as the original plaintext. We saw the use of ciphertext stealing in the case of XTS-AES to deal with partial blocks. Figure 6.13a shows the use of ciphertext stealing to modify CBC mode, called CBC-CTS.

IV P1

C1

K K K K

�

PN�2

CN�2

CN�3

• • •

�

Encrypt Encrypt Encrypt Encrypt

Encrypt Encrypt

(a) Ciphertext stealing mode

(b) Alternative method

Encrypt Encrypt

�

CN X

PN�1

�

CN�1

PN 00…0

IVP1

(bb bits)

C1(bb bits)

K K K K

�

PN�2(bb bits)

CN�2(bb bits)

CN�3

• • •

�

selectleftmost

j bits

PN�1(bb bits)

CN�1(bb bits)

�

PN(j bits)

CN(j bits)

�

Figure 6.13 Block Cipher Modes for Plaintext not a Multiple of Block Size

6.9 / key terMS, review QueStionS, and proBleMS 201

a. Explain how it works. b. Describe how to decrypt Cn - 1 and Cn. 6.12 Figure 6.13b shows an alternative to CBC-CTS for producing ciphertext of equal

length to the plaintext when the plaintext is not an integer multiple of the block size. a. Explain the algorithm. b. Explain why CBC-CTS is preferable to this approach illustrated in Figure 6.13b.

6.13 Draw a figure similar to those of Figure 6.8 for XTS-AES mode.

Programming Problems 6.14 Create software that can encrypt and decrypt in cipher block chaining mode using

one of the following ciphers: affine modulo 256, Hill modulo 256, S-DES, DES.Test data for S-DES using a binary initialization vector of 1010 1010. A binary plain-text of 0000 0001 0010 0011 encrypted with a binary key of 01111 11101 should give a binary plaintext of 1111 0100 0000 1011. Decryption should work correspondingly.

6.15 Create software that can encrypt and decrypt in 4-bit cipher feedback mode using one of the following ciphers: additive modulo 256, affine modulo 256, S-DES;

or8-bit cipher feedback mode using one of the following ciphers: 2 * 2 Hill modulo 256. Test data for S-DES using a binary initialization vector of 1010 1011. A binary plain-text of 0001 0010 0011 0100 encrypted with a binary key of 01111 11101 should give a binary plaintext of 1110 1100 1111 1010. Decryption should work correspondingly.

6.16 Create software that can encrypt and decrypt in counter mode using one of the fol-lowing ciphers: affine modulo 256, Hill modulo 256, S-DES.Test data for S-DES using a counter starting at 0000 0000. A binary plaintext of 0000 0001 0000 0010 0000 0100 encrypted with a binary key of 01111 11101 should give a binary plaintext of 0011 1000 0100 1111 0011 0010. Decryption should work correspondingly.

6.17 Implement a differential cryptanalysis attack on 3-round S-DES.

202202

7.1 Principles of Pseudorandom Number Generation

The Use of Random Numbers TRNGs, PRNGs, and PRFs PRNG Requirements Algorithm Design

7.2 Pseudorandom Number Generators

Linear Congruential Generators Blum Blum Shub Generator

7.3 Pseudorandom Number Generation Using a Block Cipher

PRNG Using Block Cipher Modes of Operation ANSI X9.17 PRNG NIST CTR_DRBG

7.4 Stream Ciphers

7.5 RC4

Initialization of S Stream Generation Strength of RC4

7.6 True Random Number Generators

Entropy Sources Comparison of PRNGs and TRNGs Skew Intel Digital Random Number Generator DRNG Hardware Architecture DRNG Logical Structure



Chapter

Pseudorandom number Generation and stream CiPhers

7.1 / PrinciPles of Pseudorandom number Generation 203

The comparatively late rise of the theory of probability shows how hard it is to grasp, and the many paradoxes show clearly that we, as humans, lack a well grounded intuition in this matter.

In probability theory there is a great deal of art in setting up the model, in solving the problem, and in applying the results back to the real world actions that will follow.

—The Art of Probability, Richard Hamming

Learning Objectives


u Explain the concepts of randomness and unpredictability with respect to random numbers.

u Understand the differences among true random number generators, pseudorandom number generators, and pseudorandom functions.

u Present an overview of requirements for pseudorandom number generators.

u Explain how a block cipher can be used to construct a pseudorandom number generator.

u Present an overview of stream ciphers and RC4.

u Explain the significance of skew.

An important cryptographic function is cryptographically strong pseudorandom num-ber generation. Pseudorandom number generators (PRNGs) are used in a variety of cryptographic and security applications. We begin the chapter with a look at the basic principles of PRNGs and contrast these with true random number generators (TRNGs).1 Next, we look at some common PRNGs, including PRNGs based on the use of a symmetric block cipher.

The chapter then moves on to the topic of symmetric stream ciphers, which are based on the use of a PRNG. The chapter next examines the most important stream cipher, RC4. Finally, we examine TRNGs.

7.1 PrinciPLes Of PseudOrandOm number generatiOn

Random numbers play an important role in the use of encryption for various net-work security applications. In this section, we provide a brief overview of the use of random numbers in cryptography and network security and then focus on the principles of pseudorandom number generation.

1A note on terminology. Some standards documents, notably NIST and ANSI, refer to a TRNG as a nondeterministic random bit generator (NRBG) and a PRNG as a deterministic random bit generator (DRBG).

204 chaPter 7 / Pseudorandom number Generation and stream ciPhers

The Use of Random Numbers

A number of network security algorithms and protocols based on cryptography make use of random binary numbers. For example,

• Key distribution and reciprocal (mutual) authentication schemes, such as those discussed in Chapters 14 and 15. In such schemes, two communicating parties cooperate by exchanging messages to distribute keys and/or authenti-cate each other. In many cases, nonces are used for handshaking to prevent replay attacks. The use of random numbers for the nonces frustrates an oppo-nent’s efforts to determine or guess the nonce, in order to repeat an obsolete transaction.

• Session key generation. We will see a number of protocols in this book where a secret key for symmetric encryption is generated for use for a particular transaction (or session) and is valid for a short period of time. This key is generally called a session key.

• Generation of keys for the RSA public-key encryption algorithm (described in Chapter 9).

• Generation of a bit stream for symmetric stream encryption (described in this chapter).

These applications give rise to two distinct and not necessarily compatible requirements for a sequence of random numbers: randomness and unpredictability.

Randomness Traditionally, the concern in the generation of a sequence of alleg-edly random numbers has been that the sequence of numbers be random in some well-defined statistical sense. The following two criteria are used to validate that a sequence of numbers is random:

• Uniform distribution: The distribution of bits in the sequence should be uni-form; that is, the frequency of occurrence of ones and zeros should be approxi-mately equal.

• Independence: No one subsequence in the sequence can be inferred from the others.

Although there are well-defined tests for determining that a sequence of bits matches a particular distribution, such as the uniform distribution, there is no such test to “prove” independence. Rather, a number of tests can be applied to demon-strate if a sequence does not exhibit independence. The general strategy is to apply a number of such tests until the confidence that independence exists is sufficiently strong. That is, if each of a number of tests fails to show that a sequence of bits is not independent, then we can have a high level of confidence that the sequence is in fact independent.

In the context of our discussion, the use of a sequence of numbers that appear statistically random often occurs in the design of algorithms related to cryptography. For example, a fundamental requirement of the RSA public-key encryption scheme discussed in Chapter 9 is the ability to generate prime numbers. In general, it is


difficult to determine if a given large number N is prime. A brute-force approach would be to divide N by every odd integer less than1N. If N is on the order, say, of 10150, which is a not uncommon occurrence in public-key cryptography, such a brute-force approach is beyond the reach of human analysts and their computers. However, a number of effective algorithms exist that test the primality of a num-ber by using a sequence of randomly chosen integers as input to relatively simple computations. If the sequence is sufficiently long (but far, far less than 210150), the primality of a number can be determined with near certainty. This type of approach, known as randomization, crops up frequently in the design of algorithms. In es-sence, if a problem is too hard or time-consuming to solve exactly, a simpler, shorter approach based on randomization is used to provide an answer with any desired level of confidence.

UnpRedictability In applications such as reciprocal authentication, session key generation, and stream ciphers, the requirement is not just that the sequence of numbers be statistically random but that the successive members of the sequence are unpredictable. With “true” random sequences, each number is statistically inde-pendent of other numbers in the sequence and therefore unpredictable. Although true random numbers are used in some applications, they have their limitations, such as inefficiency, as is discussed shortly. Thus, it is more common to implement algorithms that generate sequences of numbers that appear to be random. In this latter case, care must be taken that an opponent not be able to predict future ele-ments of the sequence on the basis of earlier elements.

TRNGs, PRNGs, and PRFs

Cryptographic applications typically make use of algorithmic techniques for ran-dom number generation. These algorithms are deterministic and therefore produce sequences of numbers that are not statistically random. However, if the algorithm is good, the resulting sequences will pass many tests of randomness. Such numbers are referred to as pseudorandom numbers.

You may be somewhat uneasy about the concept of using numbers gener-ated by a deterministic algorithm as if they were random numbers. Despite what might be called philosophical objections to such a practice, it generally works. That is, under most circumstances, pseudorandom numbers will perform as well as if they were random for a given use. The phrase “as well as” is unfortunately subjective, but the use of pseudorandom numbers is widely accepted. The same principle applies in statistical applications, in which a statistician takes a sample of a population and assumes that the results will be approximately the same as if the whole population were measured.

Figure 7.1 contrasts a true random number generator (TRNG) with two forms of pseudorandom number generators. A TRNG takes as input a source that is effectively random; the source is often referred to as an entropy source. We discuss such sources in Section 7.6. In essence, the entropy source is drawn from the physical environment of the computer and could include things such as keystroke timing patterns, disk electrical activity, mouse movements, and


instantaneous values of the system clock. The source, or combination of sources, serve as input to an algorithm that produces random binary output. The TRNG may simply involve conversion of an analog source to a binary output. The TRNG may involve additional processing to overcome any bias in the source; this is discussed in Section 7.6.

In contrast, a PRNG takes as input a fixed value, called the seed, and produces a sequence of output bits using a deterministic algorithm. Quite often, the seed is generated by a TRNG. Typically, as shown, there is some feedback path by which some of the results of the algorithm are fed back as input as additional output bits are produced. The important thing to note is that the output bit stream is determined solely by the input value or values, so that an adversary who knows the algorithm and the seed can reproduce the entire bit stream.

Figure 7.1 shows two different forms of PRNGs, based on application.

• Pseudorandom number generator: An algorithm that is used to produce an open-ended sequence of bits is referred to as a PRNG. A common application for an open-ended sequence of bits is as input to a symmetric stream cipher, as discussed in Section 7.4. Also, see Figure 3.1a.

• Pseudorandom function (PRF): A PRF is used to produced a pseudoran-dom string of bits of some fixed length. Examples are symmetric encryption keys and nonces. Typically, the PRF takes as input a seed plus some context specific values, such as a user ID or an application ID. A number of examples of PRFs will be seen throughout this book, notably in Chapters 17 and 18.

Other than the number of bits produced, there is no difference between a PRNG and a PRF. The same algorithms can be used in both applications. Both require a seed

Conversionto binary

Source oftrue

randomness

Randombit stream

(a) TRNG

TRNG = true random number generatorPRNG = pseudorandom number generatorPRF = pseudorandom function

Deterministicalgorithm

Seed

Pseudorandombit stream

(b) PRNG

Deterministicalgorithm

Seed

Pseudorandomvalue

(c) PRF

Context-specificvalues

Figure 7.1 Random and Pseudorandom Number Generators


and both must exhibit randomness and unpredictability. Further, a PRNG applica-tion may also employ context-specific input. In what follows, we make no distinction between these two applications.

PRNG Requirements

When a PRNG or PRF is used for a cryptographic application, then the basic requirement is that an adversary who does not know the seed is unable to deter-mine the pseudorandom string. For example, if the pseudorandom bit stream is used in a stream cipher, then knowledge of the pseudorandom bit stream would enable the adversary to recover the plaintext from the ciphertext. Similarly, we wish to protect the output value of a PRF. In this latter case, consider the follow-ing scenario. A 128-bit seed, together with some context-specific values, are used to generate a 128-bit secret key that is subsequently used for symmetric encryp-tion. Under normal circumstances, a 128-bit key is safe from a brute-force attack. However, if the PRF does not generate effectively random 128-bit output values, it may be possible for an adversary to narrow the possibilities and successfully use a brute force attack.

This general requirement for secrecy of the output of a PRNG or PRF leads to specific requirements in the areas of randomness, unpredictability, and the charac-teristics of the seed. We now look at these in turn.

Randomness In terms of randomness, the requirement for a PRNG is that the gen-erated bit stream appear random even though it is deterministic. There is no single test that can determine if a PRNG generates numbers that have the characteristic of randomness. The best that can be done is to apply a sequence of tests to the PRNG. If the PRNG exhibits randomness on the basis of multiple tests, then it can be assumed to satisfy the randomness requirement. NIST SP 800-22 (A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications) specifies that the tests should seek to establish the following three characteristics.

• Uniformity: At any point in the generation of a sequence of random or pseudo-random bits, the occurrence of a zero or one is equally likely, that is, the prob-ability of each is exactly 1/2. The expected number of zeros (or ones) is n/2, where n = the sequence length.

• Scalability: Any test applicable to a sequence can also be applied to subse-quences extracted at random. If a sequence is random, then any such extracted subsequence should also be random. Hence, any extracted subsequence should pass any test for randomness.

• Consistency: The behavior of a generator must be consistent across starting values (seeds). It is inadequate to test a PRNG based on the output from a single seed or an TRNG on the basis of an output produced from a single physical output.

SP 800-22 lists 15 separate tests of randomness. An understanding of these tests requires a basic knowledge of statistical analysis, so we don’t attempt a


technical description here. Instead, to give some flavor for the tests, we list three of the tests and the purpose of each test, as follows.

• Frequency test: This is the most basic test and must be included in any test suite. The purpose of this test is to determine whether the number of ones and zeros in a sequence is approximately the same as would be expected for a truly random sequence.

• Runs test: The focus of this test is the total number of runs in the sequence, where a run is an uninterrupted sequence of identical bits bounded before and after with a bit of the opposite value. The purpose of the runs test is to determine whether the number of runs of ones and zeros of various lengths is as expected for a random sequence.

• Maurer’s universal statistical test: The focus of this test is the number of bits between matching patterns (a measure that is related to the length of a com-pressed sequence). The purpose of the test is to detect whether or not the sequence can be significantly compressed without loss of information. A sig-nificantly compressible sequence is considered to be non-random.

UnpRedictability A stream of pseudorandom numbers should exhibit two forms of unpredictability:

• Forward unpredictability: If the seed is unknown, the next output bit in the sequence should be unpredictable in spite of any knowledge of previous bits in the sequence.

• Backward unpredictability: It should also not be feasible to determine the seed from knowledge of any generated values. No correlation between a seed and any value generated from that seed should be evident; each element of the sequence should appear to be the outcome of an independent random event whose probability is 1/2.

The same set of tests for randomness also provide a test of unpredictability. If the generated bit stream appears random, then it is not possible to predict some bit or bit sequence from knowledge of any previous bits. Similarly, if the bit sequence appears random, then there is no feasible way to deduce the seed based on the bit sequence. That is, a random sequence will have no correlation with a fixed value (the seed).

seed ReqUiRements For cryptographic applications, the seed that serves as input to the PRNG must be secure. Because the PRNG is a deterministic algorithm, if the adversary can deduce the seed, then the output can also be determined. Therefore, the seed must be unpredictable. In fact, the seed itself must be a random or pseudo-random number.

Typically, the seed is generated by a TRNG, as shown in Figure 7.2. This is the scheme recommended by SP800-90. The reader may wonder, if a TRNG is avail-able, why it is necessary to use a PRNG. If the application is a stream cipher, then a TRNG is not practical. The sender would need to generate a keystream of bits as long as the plaintext and then transmit the keystream and the ciphertext securely to the receiver. If a PRNG is used, the sender need only find a way to deliver the stream cipher key, which is typically 54 or 128 bits, to the receiver in a secure fashion.


Entropysource

Pseudorandomnumber generator

(PRNG)

Seed

Pseudorandombit stream

True randomnumber generator

(TRNG)

Figure 7.2 Generation of Seed Input to PRNG

Even in the case of a PRF application, in which only a limited number of bits is generated, it is generally desirable to use a TRNG to provide the seed to the PRF and use the PRF output rather than use the TRNG directly. As is explained in Section 7.6, a TRNG may produce a binary string with some bias. The PRF would have the effect of “randomizing” the output of the TRNG so as to eliminate that bias.

Finally, the mechanism used to generate true random numbers may not be able to generate bits at a rate sufficient to keep up with the application requiring the random bits.

Algorithm Design

Cryptographic PRNGs have been the subject of much research over the years, and a wide variety of algorithms have been developed. These fall roughly into two categories.

• Purpose-built algorithms: These are algorithms designed specifically and solely for the purpose of generating pseudorandom bit streams. Some of these algorithms are used for a variety of PRNG applications; several of these are described in the next section. Others are designed specifically for use in a stream cipher. The most important example of the latter is RC4, described in Section 7.5.

• Algorithms based on existing cryptographic algorithms: Cryptographic algo-rithms have the effect of randomizing input data. Indeed, this is a require-ment of such algorithms. For example, if a symmetric block cipher produced ciphertext that had certain regular patterns in it, it would aid in the process of cryptanalysis. Thus, cryptographic algorithms can serve as the core of PRNGs.


Three broad categories of cryptographic algorithms are commonly used to create PRNGs:

—Symmetric block ciphers: This approach is discussed in Section 7.3.

—Asymmetric ciphers: The number theoretic concepts used for an asym-metric cipher can also be adapted for a PRNG; this approach is examined in Chapter 10.

—Hash functions and message authentication codes: This approach is exam-ined in Chapter 12.

Any of these approaches can yield a cryptographically strong PRNG. A purpose-built algorithm may be provided by an operating system for general use. For applications that already use certain cryptographic algorithms for encryption or authentication, it makes sense to reuse the same code for the PRNG. Thus, all of these approaches are in common use.

7.2 PseudOrandOm number generatOrs

In this section, we look at two types of algorithms for PRNGs.

Linear Congruential Generators

A widely used technique for pseudorandom number generation is an algorithm first proposed by Lehmer [LEHM51], which is known as the linear congruential method. The algorithm is parameterized with four numbers, as follows:

m the modulus m 7 0a the multiplier 0 6 a 6 mc the increment 0 … c 6 mX0 the starting value, or seed 0 … X0 6 m

The sequence of random numbers {Xn} is obtained via the following iterative equation:

Xn + 1 = (aXn + c)mod m

If m, a, c, and X0 are integers, then this technique will produce a sequence of inte-gers with each integer in the range 0 … Xn 6 m.

The selection of values for a, c, and m is critical in developing a good ran-dom number generator. For example, consider a = c = 1. The sequence produced is obviously not satisfactory. Now consider the values a = 7, c = 0, m = 32, and X0 = 1. This generates the sequence {7, 17, 23, 1, 7, etc.}, which is also clearly un-satisfactory. Of the 32 possible values, only four are used; thus, the sequence is said to have a period of 4. If, instead, we change the value of a to 5, then the sequence is {5, 25, 29, 17, 21, 9, 13, 1, 5, etc.}, which increases the period to 8.

We would like m to be very large, so that there is the potential for producing a long series of distinct random numbers. A common criterion is that m be nearly equal to the maximum representable nonnegative integer for a given computer. Thus, a value of m near to or equal to 231 is typically chosen.

7.2 / Pseudorandom number Generators 211

[PARK88a] proposes three tests to be used in evaluating a random number generator:

T1: The function should be a full-period generating function. That is, the function should generate all the numbers from 0 through m - 1 before repeating.

T2: The generated sequence should appear random.

T3: The function should implement efficiently with 32-bit arithmetic.

With appropriate values of a, c, and m, these three tests can be passed. With respect to T1, it can be shown that if m is prime and c = 0, then for certain values of a the period of the generating function is m - 1, with only the value 0 missing. For 32-bit arithmetic, a convenient prime value of m is 231 - 1. Thus, the generating function becomes

Xn + 1 = (aXn) mod (231 - 1)

Of the more than 2 billion possible choices for a, only a handful of multipliers pass all three tests. One such value is a = 75 = 16807, which was originally selected for use in the IBM 360 family of computers [LEWI69]. This generator is widely used and has been subjected to a more thorough testing than any other PRNG. It is fre-quently recommended for statistical and simulation work (e.g., [JAIN91]).

The strength of the linear congruential algorithm is that if the multiplier and modulus are properly chosen, the resulting sequence of numbers will be statisti-cally indistinguishable from a sequence drawn at random (but without replacement) from the set 1, 2, c, m - 1. But there is nothing random at all about the algo-rithm, apart from the choice of the initial value X0. Once that value is chosen, the remaining numbers in the sequence follow deterministically. This has implications for cryptanalysis.

If an opponent knows that the linear congruential algorithm is being used and if the parameters are known (e.g., a = 75, c = 0, m = 231 - 1), then once a single number is discovered, all subsequent numbers are known. Even if the opponent knows only that a linear congruential algorithm is being used, knowledge of a small part of the sequence is sufficient to determine the parameters of the algorithm. Suppose that the opponent is able to determine values for X0, X1, X2, and X3. Then

X1 = (aX0 + c) mod mX2 = (aX1 + c) mod mX3 = (aX2 + c) mod m

These equations can be solved for a, c, and m.Thus, although it is nice to be able to use a good PRNG, it is desirable to make

the actual sequence used nonreproducible, so that knowledge of part of the se-quence on the part of an opponent is insufficient to determine future elements of the sequence. This goal can be achieved in a number of ways. For example, [BRIG79] suggests using an internal system clock to modify the random number stream. One way to use the clock would be to restart the sequence after every N numbers using the current clock value (mod m) as the new seed. Another way would be simply to add the current clock value to each random number (mod m).


Blum Blum Shub Generator

A popular approach to generating secure pseudorandom numbers is known as the Blum Blum Shub (BBS) generator (see Figure 7.3), named for its developers [BLUM86]. It has perhaps the strongest public proof of its cryptographic strength of any purpose-built algorithm. The procedure is as follows. First, choose two large prime numbers, p and q, that both have a remainder of 3 when divided by 4. That is,

p K q K 3(mod 4)

This notation, explained more fully in Chapter 4, simply means that (p mod 4) =(q mod 4) = 3. For example, the prime numbers 7 and 11 satisfy 7 K 11 K 3(mod 4). Let n = p * q. Next, choose a random number s, such that s is relatively prime to n; this is equivalent to saying that neither p nor q is a factor of s. Then the BBS genera-tor produces a sequence of bits Bi according to the following algorithm:

X0 = s2 mod nfor i = 1 to ∞ Xi = (Xi-1)

2 mod n Bi = Xi mod 2

Thus, the least significant bit is taken at each iteration. Table 7.1 shows an example of BBS operation. Here, n = 192649 = 383 * 503, and the seed s = 101355.

The BBS is referred to as a cryptographically secure pseudorandom bit gen-erator (CSPRBG). A CSPRBG is defined as one that passes the next-bit test, which, in turn, is defined as follows [MENE97]: A pseudorandom bit generator is said to pass the next-bit test if there is not a polynomial-time algorithm2 that, on input of the first k bits of an output sequence, can predict the (k + 1)st bit with probability

Generatex2 mod n

Select leastsignificant bit

Initializewith seed s

[0, 1]

Figure 7.3 Blum Blum Shub Block Diagram

2A polynomial-time algorithm of order k is one whose running time is bounded by a polynomial of order k.

7.3 / Pseudorandom number Generation usinG a block ciPher 213

significantly greater than 1/2. In other words, given the first k bits of the sequence, there is not a practical algorithm that can even allow you to state that the next bit will be 1 (or 0) with probability greater than 1/2. For all practical purposes, the sequence is unpredictable. The security of BBS is based on the difficulty of factoring n. That is, given n, we need to determine its two prime factors p and q.

7.3 PseudOrandOm number generatiOn using a bLOck ciPher

A popular approach to PRNG construction is to use a symmetric block cipher as the heart of the PRNG mechanism. For any block of plaintext, a symmetric block cipher produces an output block that is apparently random. That is, there are no patterns or regularities in the ciphertext that provide information that can be used to deduce the plaintext. Thus, a symmetric block cipher is a good candidate for building a pseudorandom number generator.

If an established, standardized block cipher is used, such as DES or AES, then the security characteristics of the PRNG can be established. Further, many applica-tions already make use of DES or AES, so the inclusion of the block cipher as part of the PRNG algorithm is straightforward.

PRNG Using Block Cipher Modes of Operation

Two approaches that use a block cipher to build a PNRG have gained widespread acceptance: the CTR mode and the OFB mode. The CTR mode is recommended in NIST SP 800-90, in the ANSI standard X9.82 (Random Number Generation), and in RFC 4086. The OFB mode is recommended in X9.82 and RFC 4086.

Figure 7.4 illustrates the two methods. In each case, the seed consists of two parts: the encryption key value and a value V that will be updated after each block of pseudorandom numbers is generated. Thus, for AES-128, the seed consists of a 128-bit key and a 128-bit V value. In the CTR case, the value of V is incremented by 1 after each encryption. In the case of OFB, the value of V is updated to equal the

Table 7.1 Example Operation of BBS Generator

i Xi Bi

0 20749

1 143135 12 177671 13 97048 04 89992 05 174051 16 80649 17 45663 18 69442 09 186894 010 177046 0

i Xi Bi

11 137922 012 123175 113 8630 014 114386 015 14863 116 133015 117 106065 118 45870 019 137171 120 48060 0


value of the preceding PRNG block. In both cases, pseudorandom bits are produced one block at a time (e.g., for AES, PRNG bits are generated 128 bits at a time).

The CTR algorithm for PRNG, called CTR_DRBG, can be summarized as follows.

while (len (temp) < requested_number_of_bits) do

V = (V + 1) mod 2128.

output_block = E(Key, V)

temp = temp || ouput_block

The OFB algorithm can be summarized as follows.

while (len (temp) < requested_number_of_bits) do

V = E(Key, V)

temp = temp || V

To get some idea of the performance of these two PRNGs, consider the fol-lowing short experiment. A random bit sequence of 256 bits was obtained from random.org, which uses three radios tuned between stations to pick up atmospheric noise. These 256 bits form the seed, allocated as

Key: cfb0ef3108d49cc4562d5810b0a9af60

V: 4c89af496176b728ed1e2ea8ba27f5a4

The total number of one bits in the 256-bit seed is 124, or a fraction of 0.48, which is reassuringly close to the ideal of 0.5.

For the OFB PRNG, Table 7.2 shows the first eight output blocks (1024 bits) with two rough measures of security. The second column shows the fraction of one bits in each 128-bit block. This corresponds to one of the NIST tests. The results indicate that the output is split roughly equally between zero and one bits. The third

(a) CTR mode

V

Encrypt

Pseudorandom bits

K

1

+

(b) OFB mode

V

Encrypt

Pseudorandom bits

K

Figure 7.4 PRNG Mechanisms Based on Block Ciphers


column shows the fraction of bits that match between adjacent blocks. If this num-ber differs substantially from 0.5, that suggests a correlation between blocks, which could be a security weakness. The results suggest no correlation.

Table 7.3 shows the results using the same key and V values for CTR mode. Again, the results are favorable.

ANSI X9.17 PRNG

One of the strongest (cryptographically speaking) PRNGs is specified in ANSI X9.17. A number of applications employ this technique, including financial security applications and PGP (the latter described in Chapter 19).

Figure 7.5 illustrates the algorithm, which makes use of triple DES for encryp-tion. The ingredients are as follows.

• Input: Two pseudorandom inputs drive the generator. One is a 64-bit rep-resentation of the current date and time, which is updated on each number generation. The other is a 64-bit seed value; this is initialized to some arbitrary value and is updated during the generation process.

• Keys: The generator makes use of three triple DES encryption modules. All three make use of the same pair of 56-bit keys, which must be kept secret and are used only for pseudorandom number generation.

Table 7.2 Example Results for PRNG Using OFB

Output BlockFraction of One Bits

Fraction of Bits that Match with

Preceding Block

1786f4c7ff6e291dbdfdd90ec3453176 0.57 —5e17b22b14677a4d66890f87565eae64 0.51 0.52fd18284ac82251dfb3aa62c326cd46cc 0.47 0.54c8e545198a758ef5dd86b41946389bd5 0.50 0.44fe7bae0e23019542962e2c52d215a2e3 0.47 0.4814fdf5ec99469598ae0379472803accd 0.49 0.526aeca972e5a3ef17bd1a1b775fc8b929 0.57 0.48f7e97badf359d128f00d9b4ae323db64 0.55 0.45

Table 7.3 Example Results for PRNG Using CTR

Output BlockFraction of One

Bits

Fraction of Bits that Match with

Preceding Block

1786f4c7ff6e291dbdfdd90ec3453176 0.57 —60809669a3e092a01b463472fdcae420 0.41 0.41d4e6e170b46b0573eedf88ee39bff33d 0.59 0.455f8fcfc5deca18ea246785d7fadc76f8 0.59 0.5290e63ed27bb07868c753545bdd57ee28 0.53 0.520125856fdf4a17f747c7833695c52235 0.50 0.47f4be2d179b0f2548fd748c8fc7c81990 0.51 0.481151fc48f90eebac658a3911515c3c66 0.47 0.45


• Output: The output consists of a 64-bit pseudorandom number and a 64-bit seed value.

Let us define the following quantities.

DTi Date/time value at the beginning of ith generation stageVi Seed value at the beginning of ith generation stageRi Pseudorandom number produced by the ith generation stageK1, K2 DES keys used for each stage

Then

Ri = EDE([K1, K2], [Vi ⊕ EDE([K1, K2], DTi)])Vi+ 1 = EDE([K1, K2], [Ri ⊕ EDE([K1, K2], DTi)])

where EDE([K1, K2], X ) refers to the sequence encrypt-decrypt-encrypt using two-key triple DES to encrypt X.

Several factors contribute to the cryptographic strength of this method. The technique involves a 112-bit key and three EDE encryptions for a total of nine DES encryptions. The scheme is driven by two independent inputs, the date and time value, and a seed produced by the generator that is distinct from the pseudorandom number produced by the generator. Thus, the amount of material that must be com-promised by an opponent appears to be overwhelming. Even if a pseudorandom number Ri were compromised, it would be impossible to deduce the Vi+ 1 from the Ri, because an additional EDE operation is used to produce the Vi+ 1.

NIST CTR_DRBG

We now look more closely at the details of the PRNG defined in NIST SP 800-90 based on the CTR mode of operation. The PRNG is referred to as CTR_DRBG (counter mode–deterministic random bit generator). CTR_DRBG is widely imple-mented and is part of the hardware random number generator implemented on all recent Intel processor chips (discussed in Section 7.6).

EDE

EDE

EDE

K1, K2

DTi

Vi

Vi�1

Ri

Figure 7.5 ANSI X9.17 Pseudorandom Number Generator


The DRBG assumes that an entropy source is available to provide random bits. Typically, the entropy source will be a TRNG based on some physical source. Other sources are possible if they meet the required entropy measure of the appli-cation. Entropy is an information theoretic concept that measures unpredictability, or randomness; see Appendix F for details. The encryption algorithm used in the DRBG may be 3DES with three keys or AES with a key size of 128, 192, or 256 bits.

Four parameters are associated with the algorithm:

• Output block length (outlen): Length of the output block of the encryption algorithm.

• Key length (keylen): Length of the encryption key.

• Seed length (seedlen): The seed is a string of bits that is used as input to a DRBG mechanism. The seed will determine a portion of the internal state of the DRBG, and its entropy must be sufficient to support the security strength of the DRBG. seedlen = outlen + keylen.

• Reseed interval (reseed_interval): Length of the encryption key. It is the maxi-mum number of output blocks generated before updating the algorithm with a new seed.

Table 7.4 lists the values specified in SP 800-90 for these parameters.

initialize Figure 7.6 shows the two principal functions that comprise CTR_DRBG. We first consider how CTR_DRBG is initialized, using the initialize and update function (Figure 7.6a). Recall that the CTR block cipher mode requires both an encryption key K and an initial counter value, referred to in SP 800-90 as the coun-ter V. The combination of K and V is referred to as the seed. To start the DRGB operation, initial values for K and V are needed, and can be chosen arbitrarily. As an example, the Intel Digital Random Number Generator, discussed in Section 7.6, uses the values K = 0 and V = 0. These values are used as parameters for the CTR mode of operation to produce at least seedlen bits. In addition, exactly seedlen bits must be supplied from what is referred to as an entropy source. Typically, the en-tropy source would be some form of TRNG.

With these inputs, the CTR mode of encryption is iterated to produce a sequence of output blocks, with V incremented by 1 after each encryption. The pro-cess continues until at least seedlen bits have been generated. The leftmost seedlen bits of output are then XORed with the seedlen entropy bits to produce a new seed. In turn, the leftmost keylen bits of the seed form the new key and the rightmost outlen bits of the seed form the new counter value V.

Table 7.4 CTR_DRBG Parameters

3DES AES-128 AES-192 AES-256

outlen 64 128 128 128keylen 168 128 192 256seedlen 232 256 320 384reseed_interval …232 …248 …248 …248


GeneRate Once values of Key and V are obtained, the DRBG enters the generate phase and is able to generate pseudorandom bits, one output block at a time (Figure 7.6b). The encryption function is iterated to generate the number of pseudorandom bits desired. Each iteration uses the same encryption key. The counter value V is incre-mented by 1 for each iteration.

Update To enhance security, the number of bits generated by any PRNG should be limited. CTR_DRGB uses the parameter reseed_interval to set that limit. During the generate phase, a reseed counter is initialized to 1 and then incremented with each iteration (each production of an output block). When the reseed counter

Encrypt

Iterate1

V

Entropysource

1sttime

+

Key

B0

Key

(a) Initialize and update function

(b) Generate function

Bi

V

Key V

Encrypt

Iterate1

+

Figure 7.6 CTR_DRBG Functions

7.4 / stream ciPhers 219

reaches reseed_interval, the update function is invoked (Figure 7.6a). The update function is the same as the initialize function. In the update case, the Key and V val-ues last used by the generate function serve as the input parameters to the update function. The update function takes seedlen new bits from an entropy source and produces a new seed (Key, V). The generate function can then resume production of pseudorandom bits. Note that the result of the update function is to change both the Key and V values used by the generate function.

7.4 stream ciPhers

A typical stream cipher encrypts plaintext one byte at a time, although a stream cipher may be designed to operate on one bit at a time or on units larger than a byte at a time. Figure 7.7 is a representative diagram of stream cipher structure. In this structure, a key is input to a pseudorandom bit generator that produces a stream of 8-bit numbers that are apparently random. The output of the generator, called a keystream, is combined one byte at a time with the plaintext stream using the bitwise exclusive-OR (XOR) operation. For example, if the next byte generated by the generator is 01101100 and the next plaintext byte is 11001100, then the resulting ciphertext byte is

11001100 plaintext⊕ 01101100 key stream

10100000 ciphertext

Decryption requires the use of the same pseudorandom sequence:

10100000 ciphertext⊕ 01101100 key stream

11001100 plaintext

Pseudorandom bytegenerator

(key stream generator)

Plaintextbyte stream

M

KeyK

KeyK

kPlaintext

byte streamM

Ciphertextbyte stream

CENCRYPTION

Pseudorandom bytegenerator

(key stream generator)

DECRYPTION

k

Figure 7.7 Stream Cipher Diagram


The stream cipher is similar to the one-time pad discussed in Chapter 2. The difference is that a one-time pad uses a genuine random number stream, whereas a stream cipher uses a pseudorandom number stream.

[KUMA97] lists the following important design considerations for a stream cipher.

1. The encryption sequence should have a large period. A pseudorandom num-ber generator uses a function that produces a deterministic stream of bits that eventually repeats. The longer the period of repeat the more difficult it will be to do cryptanalysis. This is essentially the same consideration that was dis-cussed with reference to the Vigenère cipher, namely that the longer the key-word the more difficult the cryptanalysis.

2. The keystream should approximate the properties of a true random number stream as close as possible. For example, there should be an approximately equal number of 1s and 0s. If the keystream is treated as a stream of bytes, then all of the 256 possible byte values should appear approximately equally often. The more random-appearing the keystream is, the more randomized the ciphertext is, making cryptanalysis more difficult.

3. Note from Figure 7.7 that the output of the pseudorandom number genera-tor is conditioned on the value of the input key. To guard against brute-force attacks, the key needs to be sufficiently long. The same considerations that apply to block ciphers are valid here. Thus, with current technology, a key length of at least 128 bits is desirable.

With a properly designed pseudorandom number generator, a stream cipher can be as secure as a block cipher of comparable key length. A potential advantage of a stream cipher is that stream ciphers that do not use block ciphers as a building block are typically faster and use far less code than do block ciphers. The example in this chapter, RC4, can be implemented in just a few lines of code. In recent years, this advantage has diminished with the introduction of AES, which is quite efficient in software. Furthermore, hardware acceleration techniques are now available for AES. For example, the Intel AES Instruction Set has machine instructions for one round of encryption and decryption and key generation. Using the hardware in-structions results in speedups of about an order of magnitude compared to pure software implementations [XU10].

One advantage of a block cipher is that you can reuse keys. In contrast, if two plaintexts are encrypted with the same key using a stream cipher, then cryptanalysis is often quite simple [DAWS96]. If the two ciphertext streams are XORed together, the result is the XOR of the original plaintexts. If the plaintexts are text strings, credit card numbers, or other byte streams with known properties, then cryptanaly-sis may be successful.

For applications that require encryption/decryption of a stream of data, such as over a data communications channel or a browser/Web link, a stream cipher might be the better alternative. For applications that deal with blocks of data, such as file transfer, e-mail, and database, block ciphers may be more appropriate. However, either type of cipher can be used in virtually any application.

A stream cipher can be constructed with any cryptographically strong PRNG, such as the ones discussed in Sections 7.2 and 7.3. In the next section, we look at a stream cipher that uses a PRNG designed specifically for the stream cipher.

7.5 / rc4 221

7.5 rc4

RC4 is a stream cipher designed in 1987 by Ron Rivest for RSA Security. It is a vari-able key size stream cipher with byte-oriented operations. The algorithm is based on the use of a random permutation. Analysis shows that the period of the cipher is overwhelmingly likely to be greater than 10100 [ROBS95a]. Eight to sixteen ma-chine operations are required per output byte, and the cipher can be expected to run very quickly in software. RC4 is used in the Secure Sockets Layer/Transport Layer Security (SSL/TLS) standards that have been defined for communication be-tween Web browsers and servers. It is also used in the Wired Equivalent Privacy (WEP) protocol and the newer WiFi Protected Access (WPA) protocol that are part of the IEEE 802.11 wireless LAN standard. RC4 was kept as a trade secret by RSA Security. In September 1994, the RC4 algorithm was anonymously posted on the Internet on the Cypherpunks anonymous remailers list.

The RC4 algorithm is remarkably simple and quite easy to explain. A variable-length key of from 1 to 256 bytes (8 to 2048 bits) is used to initialize a 256-byte state vector S, with elements S[0], S[1], c, S[255]. At all times, S contains a permutation of all 8-bit numbers from 0 through 255. For encryption and decryption, a byte k (see Figure 7.7) is generated from S by selecting one of the 255 entries in a systematic fashion. As each value of k is generated, the entries in S are once again permuted.

Initialization of S

To begin, the entries of S are set equal to the values from 0 through 255 in ascend-ing order; that is, S[0] = 0, S[1] = 1, c, S[255] = 255 . A temporary vector, T, is also created. If the length of the key K is 256 bytes, then K is transferred to T. Otherwise, for a key of length keylen bytes, the first keylen elements of T are copied from K, and then K is repeated as many times as necessary to fill out T. These pre-liminary operations can be summarized as

/* Initialization */

for i = 0 to 255 do

S[i] = i;

T[i] = K[i mod keylen];

Next we use T to produce the initial permutation of S. This involves starting with S[0] and going through to S[255], and for each S[i], swapping S[i] with another byte in S according to a scheme dictated by T[i]:

/* Initial Permutation of S */

j = 0;

for i = 0 to 255 do

j = (j + S[i] + T[i]) mod 256;

Swap (S[i], S[j]);

Because the only operation on S is a swap, the only effect is a permutation. S still contains all the numbers from 0 through 255.

222 CHaPter 7 / PseUDoranDoM nUMBer Generation anD streaM CiPHers

Stream Generation

Once the S vector is initialized, the input key is no longer used. Stream generation involves cycling through all the elements of S[i], and for each S[i], swapping S[i] with another byte in S according to a scheme dictated by the current configuration of S. After S[255] is reached, the process continues, starting over again at S[0]:

/* Stream Generation */

i, j = 0;

while (true)

i = (i + 1) mod 256;

j = (j + S[i]) mod 256;

Swap (S[i], S[j]);

t = (S[i] + S[j]) mod 256;

k = S[t];

To encrypt, XOR the value k with the next byte of plaintext. To decrypt, XOR the value k with the next byte of ciphertext.

Figure 7.8 illustrates the RC4 logic.

25525425343210S

T

S

(a) Initial state of S and T

(b) Initial permutation of S

Swap

T

K

T[i]

j = j + S[i] + T[i]

t = S[i] + S[j]

]j[S]i[S

Keylen

i

S

(c) Stream generation

Swap

j = j + S[i]

]t[S]j[S]i[S

k

i

Figure 7.8 RC4

7.6 / true random number Generators 223

Strength of RC4

A number of papers have been published analyzing methods of attacking RC4 (e.g., [KNUD98], [FLUH00], [MANT01]). None of these approaches is practical against RC4 with a reasonable key length, such as 128 bits. A more serious prob-lem is reported in [FLUH01]. The authors demonstrate that the WEP protocol, intended to provide confidentiality on 802.11 wireless LAN networks, is vulner-able to a particular attack approach. In essence, the problem is not with RC4 itself but the way in which keys are generated for use as input to RC4. This particular problem does not appear to be relevant to other applications using RC4 and can be remedied in WEP by changing the way in which keys are generated. This problem points out the difficulty in designing a secure system that involves both cryptographic functions and protocols that make use of them.

7.6 true randOm number generatOrs

Entropy Sources

A true random number generator (TRNG) uses a nondeterministic source to produce randomness. Most operate by measuring unpredictable natural pro-cesses, such as pulse detectors of ionizing radiation events, gas discharge tubes, and leaky capacitors. Intel has developed a commercially available chip that sam-ples thermal noise by amplifying the voltage measured across undriven resistors [JUN99]. LavaRnd is an open source project for creating truly random numbers using inexpensive cameras, open source code, and inexpensive hardware. The system uses a saturated CCD in a light-tight can as a chaotic source to produce the seed. Software processes the result into truly random numbers in a variety of formats.

RFC 4086 lists the following possible sources of randomness that, with care, easily can be used on a computer to generate true random sequences.

• Sound/video input: Many computers are built with inputs that digitize some real-world analog source, such as sound from a microphone or video input from a camera. The “input” from a sound digitizer with no source plugged in or from a camera with the lens cap on is essentially thermal noise. If the system has enough gain to detect anything, such input can provide reasonably high quality random bits.

• Disk drives: Disk drives have small random fluctuations in their rotational speed due to chaotic air turbulence [JAKO98]. The addition of low-level disk seek-time instrumentation produces a series of measurements that contain this randomness. Such data is usually highly correlated, so significant process-ing is needed. Nevertheless, experimentation a decade ago showed that, with such processing, even slow disk drives on the slower computers of that day could easily produce 100 bits a minute or more of excellent random data.

There is also an online service (random.org), which can deliver random sequences securely over the Internet.


Comparison of PRNGs and TRNGs

Table 7.5 summarizes the principal differences between PRNGs and TRNGs. PRNGs are efficient, meaning they can produce many numbers in a short time, and deterministic, meaning that a given sequence of numbers can be reproduced at a later date if the starting point in the sequence is known. Efficiency is a nice char-acteristic if your application needs many numbers, and determinism is handy if you need to replay the same sequence of numbers again at a later stage. PRNGs are typically also periodic, which means that the sequence will eventually repeat itself. While periodicity is hardly ever a desirable characteristic, modern PRNGs have a period that is so long that it can be ignored for most practical purposes.

TRNGs are generally rather inefficient compared to PRNGs, taking consid-erably longer time to produce numbers. This presents a difficulty in many applica-tions. For example, cryptography system in banking or national security might need to generate millions of random bits per second. TRNGs are also nondeterministic, meaning that a given sequence of numbers cannot be reproduced, although the same sequence may of course occur several times by chance. TRNGs have no period.

Skew

A TRNG may produce an output that is biased in some way, such as having more ones than zeros or vice versa. Various methods of modifying a bit stream to re-duce or eliminate the bias have been developed. These are referred to as deskewing algorithms. One approach to deskew is to pass the bit stream through a hash function, such as MD5 or SHA-1 (described in Chapter 11). The hash function produces an n-bit output from an input of arbitrary length. For deskewing, blocks of m input bits, with m Ú n, can be passed through the hash function. RFC 4086 recommends collect-ing input from multiple hardware sources and then mixing these using a hash function to produce random output.

Operating systems typically provide a built-in mechanism for generating ran-dom numbers. For example, Linux uses four entropy sources: mouse and keyboard activity, disk I/O operations, and specific interrupts. Bits are generated from these four sources and combined in a pooled buffer. When random bits are needed, the appropriate number of bits are read from the buffer and passed through the SHA-1 hash function [GUTT06].

Intel Digital Random Number Generator

As was mentioned, TRNGs have traditionally been used only for key generation and other applications where only a small number of random bits were required. This is because TRNGs have generally been inefficient, with a low bit rate of ran-dom bit production.

Table 7.5 Comparison of PRNGs and TRNGs

Pseudorandom Number Generators

True Random Number Generators

Efficiency Very efficient Generally inefficientDeterminism Deterministic NondeterministicPeriodicity Periodic Aperiodic

7.6 / true random number Generators 225

The first commercially available TRNG that achieves bit production rates comparable with that of PRNGs is the Intel digital random number generator (DRNG) [TAYL11], offered on new multicore chips since May 2012.

Two notable aspects of the DRNG:

1. It is implemented entirely in hardware. This provides greater security than a facil-ity that includes a software component. A hardware-only implementation should also be able to achieve greater computation speed than a software module.

2. The entire DRNG is on the same multicore chip as the processors. This elimi-nates the I/O delays found in other hardware random number generators.

dRnG HaRdwaRe aRcHitectURe Figure 7.9 shows the overall structure of the DRNG. The first stage of the DRNG generates random numbers from thermal noise. The heart of the stage consists of two inverters (NOT gates), with the output of each inverter connected to the input of the other. Such an arrangement has two stable states, with one inverter having an output of logical 1 and the other having an output of logical 0. The circuit is then configured so that both inverters are forced to have the same indeterminate state (both inputs and both outputs at logical 1) by clock pulses. Random thermal noise within the inverters soon jostles the two inverters into a mutually stable state. Additional circuitry is intended to compensate for any biases or correlations. This stage is capable, with current hardware, of generating random bits at a rate of 4 Gbps.

The output of the first stage is generated 512 bits at a time. To assure that the bit stream does not have skew or bias, a second stage of processing randomizes its input using a cryptographic function. In this case, the function is referred to as CBC-MAC or CMAC, as specified in NIST SP 800-38B. In essence, CMAC

Nondeterministicrandomnumbers

512 bits 256 bits

128 bits

HardwareAES CBC

MAC–basedconditioner

Hardware Random Number Generator

ProcessorChip

HardwareSP800-90

AES-CTR–basedPRNG

Hardwareentropysource

Nondeterministicrandom

seeds

RDRANDinstruction

Core 0

Core N–1 RDRANDinstruction

Figure 7.9 Intel Processor Chip with Random Number Generator


encrypts its input using the cipher block chaining (CBC) mode (Figure 6.4) and outputs the final block. We examine CMAC in detail in Chapters 12. The output of this stage is generated 256 bits at a time and is intended to exhibit true randomness with no skew or bias.

While the hardware’s circuitry generates random numbers from thermal noise much more quickly than its predecessors, it’s still not fast enough for some of to-day’s computing requirements. To enable the DRNG to generate random numbers

EncryptEncrypt

128 bits 128 bits

Clock

Transistor 1 Transistor 2

Inverters

Node A Node B

128 bits 128 bits

128 bits

Key V

K K K K

Encrypt Encrypt

Pseudorandombits

128 bits

101sttime

+

256 bits

EncryptEncrypt

128 bits

1

+

Hardwareentropysource

AES CBCMac-basedconditioner

AES-CTR-basedPRNG

K = 0

Figure 7.10 Intel DRNG Logical Structure

7.7 / recommended readinG 227

as quickly as software PRNG, and also maintain the high quality of the random num-bers, a third stage is added. This stage uses the 256-bit random numbers to seed a cryptographically secure PRNG that creates 128-bit numbers. From one 256-bit seed, the PRNG can output many pseudorandom numbers, exceeding the 3-Gbps rate of the entropy source. An upper bound of 511 128-bit samples can be generated per seed. The algorithm used for this stage is CTR_DRBG, described in Section 7.3.

The output of the DRNG is available to each of the cores on the chip via the RDRAND instruction. RDRAND retrieves a 16-, 32-, or 64-bit random value and makes it available in a software-accessible register.

Preliminary data from a pre-production sample on a system with a third generation Intel® Core™ family processor produced the following performance [INTE12]: up to 70 million RDRAND invocations per second, and a random data production rate of over 4 Gbps.

dRnG loGical stRUctURe Figure 7. 10 provides a simplified view of the logical flow of the Intel DRNG. As was described, the heart of the hardware entropy source is a pair of inverters that feed each other. Two transistors, driven by the same clock, force the inputs and outputs of both inverters to the logical 1 state. Because this an unstable state, thermal noise will cause the configuration to settle randomly into a stable state with either Node A at logical 1 and Node B at logical 0, or the reverse. Thus the module generates random bits at the clock rate.

The output of the entropy source is collected 512 bits at a time and used to feed to two CBC hardware implementations using AES encryption. Each imple-mentation takes two blocks of 128 bits of “plaintext” and encrypts using the CBC mode. The output of the second encryption is retained. For both CBC modules, an all-zeros key is used initially. Subsequently, the output of the PRNG stage is fed back to become the key for the conditioner stage.

The output of the conditioner stage consists of 256 bits. This block is provided as input to the update function of the PRNG stage. The update function is initial-ized with the all-zeros key and the counter value 0. The function is iterated twice to produce a 256-bit block, which is then XORed with the input from the conditioner stage. The results are used as the 128-bit key and the 128-bit seed for the generate function. The generate function produces pseudorandom bits in 128-bit blocks.


Perhaps the best treatment of PRNGs is found in [KNUT98]. An alternative to the stan-dard linear congruential algorithm, known as the linear recurrence algorithm, is explained in some detail in [BRIG79]. [ZENG91] assesses various PRNG algorithms for use in generating variable-length keys for Vernam types of ciphers.

An excellent survey of PRNGs, with an extensive bibliography, is [RITT91]. [MENE97] also provides a good discussions of secure PRNGs. Another good treatment, with an empha-sis on practical implementation issues, is RFC 4086 [EAST05]. This RFC also describes a number of deskewing techniques. [KELS98] is a good survey of secure PRNG techniques and cryptanalytic attacks on them. SP 800-90 [BARK12b] provides a useful treatment of a variety of PRNGs recommended by NIST. SP 800-22 [RUKH10] defines and discusses the 15 statistical tests of randomness recommended by NIST.


[KUMA97] contains an excellent and lengthy discussion of stream cipher design prin-ciples. Another good treatment, quite mathematical, is [RUEP92]. [ROBS95a] is an interest-ing and worthwhile examination of many design issues related to stream ciphers.

BARK12b Barker, E., and Kelsey, J. Recommendation for Random Number Generation Using Deterministic Random Bit Generators. NIST SP 800-90A, January 2012.

BRIG79 Bright, H., and Enison, R. “Quasi-Random Number Sequences from Long-Period TLP Generator with Remarks on Application to Cryptography.” Computing Surveys, December 1979.

EAST05 Eastlake, D.; Schiller, J.; and Crocker, S. Randomness Requirements for Security. RFC 4086, June 2005.

KELS98 Kelsey, J.; Schneier, B.; and Hall, C. “Cryptanalytic Attacks on Pseudorandom Number Generators.” Proceedings, Fast Software Encryption, 1998. http://www.schneier.com/paper-prngs.html


KUMA97 Kumar, I. Cryptology. Laguna Hills, CA: Aegean Park Press, 1997.MENE97 Menezes, A.; Oorshcot, P.; and Vanstone, S. Handbook of Applied

Cryptography. Boca Raton, FL: CRC Press, 1997.RITT91 Ritter, T. “The Efficient Generation of Cryptographic Confusion Sequences.”

Cryptologia, Vol. 15, No. 2, 1991. www.ciphersbyritter.com/ARTS/CRNG2ART.HTMROBS95a Robshaw, M. Stream Ciphers. RSA Laboratories Technical Report TR-701,

July 1995.RUEP92 Rueppel, T. “Stream Ciphers.” In [SIMM92].RUKH10 Rukhin, A., et al. A Statistical Test Suite for Random and Pseudorandom

Number Generators for Cryptographic Applications. NIST SP 800-22, April 2010.SIMM92 Simmons, G., ed. Contemporary Cryptology: The Science of Information

Integrity. Piscataway, NJ: IEEE Press, 1992.ZENG91 Zeng, K.; Yang, C.; Wei, D.; and Rao, T. “Pseudorandom Bit Generators in

Stream-Cipher Cryptography.” Computer, February 1991.

7.8 key terms, review QuestiOns, and PrObLems

Key Terms

backward unpredictabilityBlum Blum Shub generatordeskewingentropy sourceforward unpredictabilitykeystreamlinear congruential generator

pseudorandom function (PRF)

pseudorandom number generator (PRNG)

randomnessRC4seed

stream cipherskewtrue random number

generator (TRNG)unpredictability

http://www.schneier.com/paper-prngs.html


www.ciphersbyritter.com/ARTS/CRNG2ART.HTM

7.8 / key terms, review Questions, and Problems 229

Review Questions 7.1 What is the difference between statistical randomness and unpredictability? 7.2 List important design considerations for a stream cipher. 7.3 Why is it not desirable to reuse a stream cipher key? 7.4 What primitive operations are used in RC4?

Problems 7.1 If we take the linear congruential algorithm with an additive component of 0,

Xn + 1 = (aXn) mod m

Then it can be shown that if m is prime and if a given value of a produces the maxi-mum period of m - 1, then ak will also produce the maximum period, provided that k is less than m and that k and m - 1 are relatively prime. Demonstrate this by using X0 = 1 and m = 31 and producing the sequences for ak = 3, 32, 33, and 34.

7.2 a. What is the maximum period obtainable from the following generator?

Xn + 1 = (aXn) mod 24

b. What should be the value of a? c. What restrictions are required on the seed? 7.3 You may wonder why the modulus m = 231 - 1 was chosen for the linear congruen-

tial method instead of simply 231, because this latter number can be represented with no additional bits and the mod operation should be easier to perform. In general, the modulus 2k - 1 is preferable to 2k. Why is this so?

7.4 With the linear congruential algorithm, a choice of parameters that provides a full period does not necessarily provide a good randomization. For example, consider the following two generators:

Xn + 1 = (6Xn) mod 13

Xn + 1 = (7Xn) mod 13

Write out the two sequences to show that both are full period. Which one appears more random to you?

7.5 In any use of pseudorandom numbers, whether for encryption, simulation, or statisti-cal design, it is dangerous to trust blindly the random number generator that happens to be available in your computer’s system library. [PARK88] found that many con-temporary textbooks and programming packages make use of flawed algorithms for pseudorandom number generation. This exercise will enable you to test your system.

The test is based on a theorem attributed to Ernesto Cesaro (see [KNUT98] for a proof), which states the following: Given two randomly chosen integers, x and y, the probability that gcd(x, y) = 1 is 6/p2. Use this theorem in a program to deter-mine statistically the value of p. The main program should call three subprograms: the random number generator from the system library to generate the random integers; a subprogram to calculate the greatest common divisor of two integers using Euclid’s Algorithm; and a subprogram that calculates square roots. If these latter two programs are not available, you will have to write them as well. The main program should loop through a large number of random numbers to give an estimate of the aforementioned probability. From this, it is a simple matter to solve for your estimate of p.

If the result is close to 3.14, congratulations! If not, then the result is probably low, usually a value of around 2.7. Why would such an inferior result be obtained?


7.6 Suppose you have a true random bit generator where each bit in the generated stream has the same probability of being a 0 or 1 as any other bit in the stream and that the bits are not correlated; that is the bits are generated from identical independent dis-tribution. However, the bit stream is biased. The probability of a 1 is 0.5 + 0 and the probability of a 0 is n, where 0 6 0 6 0.5. A simple deskewing algorithm is as fol-lows: Examine the bit stream as a sequence of nonoverlapping pairs. Discard all 00 and 11 pairs. Replace each 01 pair with 0 and each 10 pair with 1.

a. What is the probability of occurrence of each pair in the original sequence? b. What is the probability of occurrence of 0 and 1 in the modified sequence? c. What is the expected number of input bits to produce x output bits? d. Suppose that the algorithm uses overlapping successive bit pairs instead of non-

overlapping successive bit pairs. That is, the first output bit is based on input bits 1 and 2, the second output bit is based on input bits 2 and 3, and so on. What can you say about the output bit stream?

7.7 Another approach to deskewing is to consider the bit stream as a sequence of non-overlapping groups of n bits each and output the parity of each group. That is, if a group contains an odd number of ones, the output is 1; otherwise the output is 0.

a. Express this operation in terms of a basic Boolean function. b. Assume, as in the preceding problem, that the probability of a 1 is 0.5 + 0. If each

group consists of 2 bits, what is the probability of an output of 1? c. If each group consists of 4 bits, what is the probability of an output of 1? d. Generalize the result to find the probability of an output of 1 for input groups of

n bits. 7.8 What RC4 key value will leave S unchanged during initialization? That is, after the

initial permutation of S, the entries of S will be equal to the values from 0 through 255 in ascending order.

7.9 RC4 has a secret internal state which is a permutation of all the possible values of the vector S and the two indices i and j.

a. Using a straightforward scheme to store the internal state, how many bits are used?

b. Suppose we think of it from the point of view of how much information is repre-sented by the state. In that case, we need to determine how may different states there are, then take the log to base 2 to find out how many bits of information this represents. Using this approach, how many bits would be needed to represent the state?

7.10 Alice and Bob agree to communicate privately via email using a scheme based on RC4, but they want to avoid using a new secret key for each transmission. Alice and Bob privately agree on a 128-bit key k. To encrypt a message m, consisting of a string of bits, the following procedure is used.

1. Choose a random 80-bit value v 2. Generate the ciphertext c = RC4(v }k) ⊕ m 3. Send the bit string (v } c) a. Suppose Alice uses this procedure to send a message m to Bob. Describe how Bob

can recover the message m from (v } c) using k. b. If an adversary observes several values (v1 } c1), (v2 } c2), . . . transmitted between

Alice and Bob, how can he/she determine when the same key stream has been used to encrypt two messages?

c. Approximately how many messages can Alice expect to send before the same key stream will be used twice? Use the result from the birthday paradox described in Appendix 11A [Equation (11.7)].

d. What does this imply about the lifetime of the key k (i.e., the number of messages that can be encrypted using k)?

Part 2: asymmetric ciPhers

Chapter

More NuMber Theory8.1 Prime Numbers

8.2 Fermat’s and Euler’s Theorems

Fermat’s TheoremEuler’s Totient FunctionEuler’s Theorem

8.3 Testing for Primality

Miller-Rabin AlgorithmA Deterministic Primality AlgorithmDistribution of Primes

8.4 The Chinese Remainder Theorem

8.5 Discrete Logarithms

The Powers of an Integer, Modulo nLogarithms for Modular ArithmeticCalculation of Discrete Logarithms



231

232 Chapter 8 / More NuMber theory

The Devil said to Daniel Webster: “Set me a task I can’t carry out, and I’ll give you anything in the world you ask for.”

Daniel Webster: “Fair enough. Prove that for n greater than 2, the equation an + bn = cn has no non-trivial solution in the integers.”They agreed on a three-day period for the labor, and the Devil disappeared.

At the end of three days, the Devil presented himself, haggard, jumpy, biting his lip. Daniel Webster said to him, “Well, how did you do at my task? Did you prove the theorem?”

“Eh? No … no, I haven’t proved it.”“Then I can have whatever I ask for? Money? The Presidency?”“What? Oh, that—of course. But listen! If we could just prove the following

two lemmas—”

—The Mathematical Magpie, Clifton Fadiman

Learning Objectives


u Discuss key concepts relating to prime numbers.

u Understand Fermat’s theorem.

u Understand Euler’s theorem.

u Define Euler’s totient function.

u Make a presentation on the topic of testing for primality.

u Explain the Chinese remainder theorem.

u Define discrete logarithms.

A number of concepts from number theory are essential in the design of public-key cryptographic algorithms. This chapter provides an overview of the concepts referred to in other chapters. The reader familiar with these topics can safely skip this chapter. The reader should also review Sections 4.1 through 4.3 before proceeding with this chapter.

As with Chapter 4, this chapter includes a number of examples, each of which is highlighted in a shaded box.

8.1 Prime numbers1

A central concern of number theory is the study of prime numbers. Indeed, whole books have been written on the subject (e.g., [CRAN01], [RIBE96]). In this section, we provide an overview relevant to the concerns of this book.

1In this section, unless otherwise noted, we deal only with the nonnegative integers. The use of negative integers would introduce no essential differences.

8.1 / priMe NuMbers 233

An integer p 7 1 is a prime number if and only if its only divisors2 are {1 and {p. Prime numbers play a critical role in number theory and in the techniques discussed in this chapter. Table 8.1 shows the primes less than 2000. Note the way the primes are distributed. In particular, note the number of primes in each range of 100 numbers.

Any integer a 7 1 can be factored in a unique way as

a = p1a1 * p2

a2 * c * ptat (8.1)

where p1 6 p2 6 c 6 pt are prime numbers and where each ai is a positive inte-ger. This is known as the fundamental theorem of arithmetic; a proof can be found in any text on number theory.

91 = 7 * 13

3600 = 24 * 32 * 52

11011 = 7 * 112 * 13

It is useful for what follows to express this another way. If P is the set of all prime numbers, then any positive integer a can be written uniquely in the following form:

a = qp∈P

pap where each ap Ú 0

The right-hand side is the product over all possible prime numbers p; for any par-ticular value of a, most of the exponents ap will be 0.

The value of any given positive integer can be specified by simply listing all the nonzero exponents in the foregoing formulation.

The integer 12 is represented by {a2 = 2, a3 = 1}.



Multiplication of two numbers is equivalent to adding the corresponding ex-

ponents. Given a = qp∈P

pap, b = qp∈P

pbp. Define k = ab. We know that the integer

k can be expressed as the product of powers of primes: k = qp∈P

pkp. It follows that kp = ap + bp for all p ∈ P.

2Recall from Chapter 4 that integer a is said to be a divisor of integer b if there is no remainder on division. Equivalently, we say that a divides b.

234

Table 8.1 Primes Under 2000

2 101 211 307 401 503 601 701 809 907 1009 1103 1201 1301 1409 1511 1601 1709 1801 19013 103 223 311 409 509 607 709 811 911 1013 1109 1213 1303 1423 1523 1607 1721 1811 19075 107 227 313 419 521 613 719 821 919 1019 1117 1217 1307 1427 1531 1609 1723 1823 19137 109 229 317 421 523 617 727 823 929 1021 1123 1223 1319 1429 1543 1613 1733 1831 1931

11 113 233 331 431 541 619 733 827 937 1031 1129 1229 1321 1433 1549 1619 1741 1847 193313 127 239 337 433 547 631 739 829 941 1033 1151 1231 1327 1439 1553 1621 1747 1861 194917 131 241 347 439 557 641 743 839 947 1039 1153 1237 1361 1447 1559 1627 1753 1867 195119 137 251 349 443 563 643 751 853 953 1049 1163 1249 1367 1451 1567 1637 1759 1871 197323 139 257 353 449 569 647 757 857 967 1051 1171 1259 1373 1453 1571 1657 1777 1873 197929 149 263 359 457 571 653 761 859 971 1061 1181 1277 1381 1459 1579 1663 1783 1877 198731 151 269 367 461 577 659 769 863 977 1063 1187 1279 1399 1471 1583 1667 1787 1879 199337 157 271 373 463 587 661 773 877 983 1069 1193 1283 1481 1597 1669 1789 1889 1997

41 163 277 379 467 593 673 787 881 991 1087 1289 1483 1693 1999

43 167 281 383 479 599 677 797 883 997 1091 1291 1487 1697

47 173 283 389 487 683 887 1093 1297 1489 1699

53 179 293 397 491 691 1097 1493

59 181 499 1499

61 191

67 193

71 197

73 199

79

83

89

97

k = 12 * 18 = (22 * 3) * (2 * 32) = 216

k2 = 2 + 1 = 3; k3 = 1 + 2 = 3

216 = 23 * 33 = 8 * 27

What does it mean, in terms of the prime factors of a and b, to say that a divides b? Any integer of the form pn can be divided only by an integer that is of a lesser or equal power of the same prime number, pj with j … n. Thus, we can say the following.

Given

a = qp∈P

pap, b = qp∈P

pbp

If a | b, then ap … bp for all p.

a = 12; b = 36; 12 | 36

12 = 22 * 3; 36 = 22 * 32

a2 = 2 = b2

a3 = 1 … 2 = b3

Thus, the inequality ap … bp is satisfied for all prime numbers.

It is easy to determine the greatest common divisor3 of two positive integers if we express each integer as the product of primes.

3Recall from Chapter 4 that the greatest common divisor of integers a and b, expressed (gcd a, b), is an integer c that divides both a and b without remainder and that any divisor of a and b is a divisor of c.

300 = 22 * 31 * 52

18 = 21 * 32

gcd(18, 300) = 21 * 31 * 50 = 6

The following relationship always holds:

If k = gcd(a, b), then kp = min(ap, bp) for all p.

Determining the prime factors of a large number is no easy task, so the pre-ceding relationship does not directly lead to a practical method of calculating the greatest common divisor.

8.1 / priMe NuMbers 235


8.2 Fermat’s and euLer’s theOrems

Two theorems that play important roles in public-key cryptography are Fermat’s theorem and Euler’s theorem.

Fermat’s Theorem4

Fermat’s theorem states the following: If p is prime and a is a positive integer not divisible by p, then

ap - 1 K 1 (mod p) (8.2)

Proof: Consider the set of positive integers less than p: {1, 2, c , p - 1} and multiply each element by a, modulo p, to get the set X = {a mod p, 2a mod p, c , (p - 1)a mod p}. None of the elements of X is equal to zero because p does not divide a. Furthermore, no two of the integers in X are equal. To see this, assume that ja K ka (mod p)), where 1 … j 6 k … p - 1. Because a is relatively prime5 to p, we can eliminate a from both sides of the equation [see Equation (4.3)] resulting in j K k (mod p). This last equality is impossible, because j and k are both positive integers less than p. Therefore, we know that the (p - 1) elements of X are all positive integers with no two elements equal. We can conclude the X consists of the set of integers {1, 2, c , p - 1} in some order. Multiplying the numbers in both sets (p and X) and taking the result mod p yields

a * 2a * c * (p - 1)a K [(1 * 2 * c * (p - 1)] (mod p)

ap - 1(p - 1)! K (p - 1)! (mod p)

We can cancel the (p - 1)! term because it is relatively prime to p [see Equation (4.5)]. This yields Equation (8.2), which completes the proof.

4This is sometimes referred to as Fermat’s little theorem.5Recall from Chapter 4 that two numbers are relatively prime if they have no prime factors in common; that is, their only common divisor is 1. This is equivalent to saying that two numbers are relatively prime if their greatest common divisor is 1.

a = 7, p = 19

72 = 49 K 11 (mod 19)

74 K 121 K 7 (mod 19)

78 K 49 K 11 (mod 19)

716 K 121 K 7 (mod 19)

ap - 1 = 718 = 716 * 72 K 7 * 11 K 1 (mod 19)

An alternative form of Fermat’s theorem is also useful: If p is prime and a is a positive integer, then

ap K a(mod p) (8.3)

8.2 / FerMat’s aNd euler’s theoreMs 237

Note that the first form of the theorem [Equation (8.2)] requires that a be relatively prime to p, but this form does not.

p = 5, a = 3 ap = 35 = 243 K 3(mod 5) = a(mod p) p = 5, a = 10 ap = 105 = 100000 K 10(mod 5) K 0(mod 5) = a(mod p)

Euler’s Totient Function

Before presenting Euler’s theorem, we need to introduce an important quantity in number theory, referred to as Euler’s totient function, written f(n), and defined as the number of positive integers less than n and relatively prime to n. By convention, f(1) = 1.

Determine f(37) anD f(35).

Because 37 is prime, all of the positive integers from 1 through 36 are relatively prime to 37. Thus f(37) = 36.

To determine f(35), we list all of the positive integers less than 35 that are rela-tively prime to it:

1, 2, 3, 4, 6, 8, 9, 11, 12, 13, 16, 17, 18

19, 22, 23, 24, 26, 27, 29, 31, 32, 33, 34

There are 24 numbers on the list, so f(35) = 24.

Table 8.2 lists the first 30 values of f(n). The value f(1) is without meaning but is defined to have the value 1.

It should be clear that, for a prime number p,

f(p) = p - 1

Now suppose that we have two prime numbers p and q with p ≠ q. Then we can show that, for n = pq,

f(n) = f(pq) = f(p) * f(q) = (p - 1) * (q - 1)

To see that f(n) = f(p) * f(q), consider that the set of positive integers less that n is the set {1, c, (pq - 1)}. The integers in this set that are not relatively prime to n are the set {p, 2p, c, (q - 1)p} and the set {q, 2q, c, (p - 1)q}. Accordingly,

f(n) = (pq - 1) - [(q - 1) + (p - 1)]

= pq - (p + q) + 1

= (p - 1) * (q - 1)

= f(p) * f(q)


Euler’s Theorem

Euler’s theorem states that for every a and n that are relatively prime:

af(n) K 1(mod n) (8.4)

Proof: Equation (8.4) is true if n is prime, because in that case, f(n) = (n - 1) and Fermat’s theorem holds. However, it also holds for any integer n. Recall that f(n) is the number of positive integers less than n that are relatively prime to n. Consider the set of such integers, labeled as

R = {x1, x2, c , xf(n)}

That is, each element xi of R is a unique positive integer less than n with gcd(xi, n) = 1. Now multiply each element by a, modulo n:

S = {(ax1 mod n), (ax2 mod n), c , (axf(n) mod n)}

The set S is a permutation6 of R, by the following line of reasoning:

1. Because a is relatively prime to n and xi is relatively prime to n, axi must also be relatively prime to n. Thus, all the members of S are integers that are less than n and that are relatively prime to n.

2. There are no duplicates in S. Refer to Equation (4.5). If axi mod n= axj mod n, then xi = xj.

Therefore,

qf(n)

i = 1(axi mod n) = q

f(n)

i = 1xi

qf(n)

i = 1axi K q

f(n)

i = 1xi (mod n)

Table 8.2 Some Values of Euler’s Totient Function f(n)

n f(n)

1 12 13 24 25 46 27 68 49 610 4

n f(n)

11 1012 413 1214 615 816 817 1618 619 1820 8

n f(n)

21 1222 1023 2224 825 2026 1227 1828 1229 2830 8

6Recall from Chapter 2 that a permutation of a finite set of elements S is an ordered sequence of all the elements of S, with each element appearing exactly once.

f(21) = f(3) * f(7) = (3 - 1) * (7 - 1) = 2 * 6 = 12

where the 12 integers are {1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20}.

8.3 / testiNg For priMality 239

af(n) * £ qf(n)

i = 1xi § K q

f(n)

i = 1xi (mod n)

af(n) K 1 (mod n)

which completes the proof. This is the same line of reasoning applied to the proof of Fermat’s theorem.

7Also referred to in the literature as the Rabin-Miller algorithm, or the Rabin-Miller test, or the Miller-Rabin test.

a = 3; n = 10; f(10) = 4 af(n) = 34 = 81 = 1 (mod 10) = 1 (mod n)

a = 2; n = 11; f(11) = 10 af(n) = 210 = 1024 = 1 (mod 11) = 1 (mod n)

As is the case for Fermat’s theorem, an alternative form of the theorem is also useful:

af(n) + 1 K a (mod n) (8.5)

Again, similar to the case with Fermat’s theorem, the first form of Euler’s theorem [Equation (8.4)] requires that a be relatively prime to n, but this form does not.

8.3 testing FOr PrimaLity

For many cryptographic algorithms, it is necessary to select one or more very large prime numbers at random. Thus, we are faced with the task of determining whether a given large number is prime. There is no simple yet efficient means of accomplish-ing this task.

In this section, we present one attractive and popular algorithm. You may be surprised to learn that this algorithm yields a number that is not necessarily a prime. However, the algorithm can yield a number that is almost certainly a prime. This will be explained presently. We also make reference to a deterministic algorithm for finding primes. The section closes with a discussion concerning the distribution of primes.

Miller-Rabin Algorithm7

The algorithm due to Miller and Rabin [MILL75, RABI80] is typically used to test a large number for primality. Before explaining the algorithm, we need some back-ground. First, any positive odd integer n Ú 3 can be expressed as

n - 1 = 2kq with k 7 0, q odd

To see this, note that n - 1 is an even integer. Then, divide (n - 1) by 2 until the result is an odd number q, for a total of k divisions. If n is expressed as a binary number, then the result is achieved by shifting the number to the right until the


rightmost digit is a 1, for a total of k shifts. We now develop two properties of prime numbers that we will need.

Two ProPerTies of Prime Numbers The first property is stated as follows: If p is prime and a is a positive integer less than p, then a2 mod p = 1 if and only if either a mod p = 1 or a mod p = -1 mod p = p - 1. By the rules of modular arithme-tic (a mod p) (a mod p) = a2 mod p. Thus, if either a mod p = 1 or a mod p = -1, then a2 mod p = 1. Conversely, if a2 mod p = 1, then (a mod p)2 = 1, which is true only for a mod p = 1 or a mod p = -1.

The second property is stated as follows: Let p be a prime number greater than 2. We can then write p - 1 = 2kq with k 7 0, q odd. Let a be any integer in the range 1 6 a 6 p - 1. Then one of the two following conditions is true.

1. aq is congruent to 1 modulo p. That is, aq mod p = 1, or equivalently, aq K 1(mod p).

2. One of the numbers aq, a2q, a4q, c, a2k - 1q is congruent to -1 modulo p. That is, there is some number j in the range (1 … j … k) such that a2 j - 1q mod p = -1 mod p = p - 1 or equivalently, a2j - 1q K -1 (mod p).

Proof: Fermat’s theorem [Equation (8.2)] states that an - 1 K 1 (mod n) if n is prime. We have p - 1 = 2kq. Thus, we know that ap - 1 mod p = a2kq mod p = 1. Thus, if we look at the sequence of numbers

aq mod p, a2q mod p, a4q mod p, c, a2k - 1q mod p, a2kq mod p (8.6)

we know that the last number in the list has value 1. Further, each number in the list is the square of the previous number. Therefore, one of the following possibilities must be true.

1. The first number on the list, and therefore all subsequent numbers on the list, equals 1.

2. Some number on the list does not equal 1, but its square mod p does equal 1. By virtue of the first property of prime numbers defined above, we know that the only number that satisfies this condition is p - 1. So, in this case, the list contains an element equal to p - 1.

This completes the proof.

DeTails of The algoriThm These considerations lead to the conclusion that, if n is prime, then either the first element in the list of residues, or remainders, (aq, a2q, c, a2k - 1q, a2kq) modulo n equals 1; or some element in the list equals (n - 1); otherwise n is composite (i.e., not a prime). On the other hand, if the condition is met, that does not necessarily mean that n is prime. For example, if n = 2047 = 23 * 89, then n - 1 = 2 * 1023. We compute 21023 mod 2047 = 1, so that 2047 meets the condition but is not prime.

We can use the preceding property to devise a test for primality. The proce-dure TEST takes a candidate integer n as input and returns the result composite if n is definitely not a prime, and the result inconclusive if n may or may not be a prime.

8.3 / testiNg For priMality 241

TEST (n)1. Find integers k, q, with k 7 0, q odd, so that

(n - 1 = 2kq);2. Select a random integer a, 1 6 a 6 n - 1;3. if aqmod n = 1 then return("inconclusive");4. for j = 0 to k - 1 do5. if a2

jqmod n = n - 1 then return("inconclusive");6. return("composite");

Let us apply the test to the prime number n = 29. We have (n - 1) = 28 = 22(7) = 2kq. First, let us try a = 10. We compute 107 mod 29 = 17, which is neither 1 nor 28, so we continue the test. The next calculation finds that (107)2 mod 29 = 28, and the test returns inconclusive (i.e., 29 may be prime). Let’s try again with a = 2. We have the following calculations: 27 mod 29 = 12; 214 mod 29 = 28; and the test again returns inconclusive. If we perform the test for all integers a in the range 1 through 28, we get the same inconclusive result, which is compatible with n being a prime number.

Now let us apply the test to the composite number n = 13 * 17 = 221. Then (n - 1) = 220 = 22(55) = 2kq. Let us try a = 5. Then we have 555 mod 221 = 112, which is neither 1 nor 220 (555)2 mod 221 = 168. Because we have used all values of j (i.e., j = 0 and j = 1) in line 4 of the TEST algorithm, the test returns composite, indicating that 221 is definitely a composite number. But suppose we had selected a = 21. Then we have 2155 mod 221 = 200; (2155)2 mod 221 = 220; and the test returns inconclusive, indicating that 221 may be prime. In fact, of the 218 integers from 2 through 219, four of these will return an inconclusive result, namely 21, 47, 174, and 200.

rePeaTeD use of The miller-rabiN algoriThm How can we use the Miller-Rabin algorithm to determine with a high degree of confidence whether or not an integer is prime? It can be shown [KNUT98] that given an odd number n that is not prime and a randomly chosen integer, a with 1 6 a 6 n - 1, the probability that TEST will return inconclusive (i.e., fail to detect that n is not prime) is less than 1/4. Thus, if t different values of a are chosen, the probability that all of them will pass TEST (return inconclusive) for n is less than (1/4)t. For example, for t = 10, the probability that a nonprime number will pass all ten tests is less than 10-6. Thus, for a sufficiently large value of t, we can be confident that n is prime if Miller’s test always returns inconclusive.

This gives us a basis for determining whether an odd integer n is prime with a reasonable degree of confidence. The procedure is as follows: Repeatedly invoke TEST (n) using randomly chosen values for a. If, at any point, TEST re-turns composite, then n is determined to be nonprime. If TEST continues to return inconclusive for t tests, then for a sufficiently large value of t, assume that n is prime.


A Deterministic Primality Algorithm

Prior to 2002, there was no known method of efficiently proving the primality of very large numbers. All of the algorithms in use, including the most popular (Miller-Rabin), produced a probabilistic result. In 2002 (announced in 2002, published in 2004), Agrawal, Kayal, and Saxena [AGRA04] developed a relatively simple deterministic algorithm that efficiently determines whether a given large number is a prime. The algo-rithm, known as the AKS algorithm, does not appear to be as efficient as the Miller-Rabin algorithm. Thus far, it has not supplanted this older, probabilistic technique.

Distribution of Primes

It is worth noting how many numbers are likely to be rejected before a prime num-ber is found using the Miller-Rabin test, or any other test for primality. A result from number theory, known as the prime number theorem, states that the primes near n are spaced on the average one every ln (n) integers. Thus, on average, one would have to test on the order of ln(n) integers before a prime is found. Because all even integers can be immediately rejected, the correct figure is 0.5 ln(n). For example, if a prime on the order of magnitude of 2200 were sought, then about 0.5 ln(2200) = 69 trials would be needed to find a prime. However, this figure is just an average. In some places along the number line, primes are closely packed, and in other places there are large gaps.

8The CRT is so called because it is believed to have been discovered by the Chinese mathematician Sun-Tsu in around 100 A.D.

The two consecutive odd integers 1,000,000,000,061 and 1,000,000,000,063 are both prime. On the other hand, 1001! + 2, 1001! + 3, c, 1001! + 1000, 1001! +1001 is a sequence of 1000 consecutive composite integers.

8.4 the chinese remainder theOrem

One of the most useful results of number theory is the Chinese remainder theorem (CRT).8 In essence, the CRT says it is possible to reconstruct integers in a certain range from their residues modulo a set of pairwise relatively prime moduli.

The 10 integers in Z10, that is the integers 0 through 9, can be reconstructed from their two residues modulo 2 and 5 (the relatively prime factors of 10). Say the known residues of a decimal digit x are r2 = 0 and r5 = 3; that is, x mod 2 = 0 and x mod 5 = 3. Therefore, x is an even integer in Z10 whose remainder, on divi-sion by 5, is 3. The unique solution is x = 8.

The CRT can be stated in several ways. We present here a formulation that is most useful from the point of view of this text. An alternative formulation is explored in Problem 8.17. Let

M = qk

i = 1mi

8.4 / the ChiNese reMaiNder theoreM 243

where the mi are pairwise relatively prime; that is, gcd(mi, mj) = 1 for 1 … i, j … k, and i ≠ j. We can represent any integer A in ZM by a k-tuple whose elements are in Zmi

using the following correspondence:

A 4 (a1, a2, c, ak) (8.7)

where A ∈ ZM, ai ∈ Zmi, and ai = A mod mi for 1 … i … k. The CRT makes two

assertions.

1. The mapping of Equation (8.7) is a one-to-one correspondence (called a bijection) between ZM and the Cartesian product Zm1

* Zm2* c * Zmk

. That is, for every integer A such that 0 … A … M, there is a unique k-tuple (a1, a2, c, ak) with 0 … ai 6 mi that represents it, and for every such k-tuple (a1, a2, c, ak), there is a unique integer A in ZM.

2. Operations performed on the elements of ZM can be equivalently performed on the corresponding k-tuples by performing the operation independently in each coordinate position in the appropriate system.

Let us demonstrate the first assertion. The transformation from A to (a1, a2, c, ak), is obviously unique; that is, each ai is uniquely calculated as ai = A mod mi. Computing A from (a1, a2, c, ak) can be done as follows. Let Mi = M/mi for 1 … i … k. Note that Mi = m1 * m2 * c * mi- 1 * mi+ 1 * c* mk, so that Mi K 0 (mod mj) for all j ≠ i. Then let

ci = Mi * 1Mi-1 mod mi2 for 1 … i … k (8.8)

By the definition of Mi, it is relatively prime to mi and therefore has a unique mul-tiplicative inverse mod mi. So Equation (8.8) is well defined and produces a unique value ci. We can now compute

A K £ ak

i- 1aici≥(mod M) (8.9)

To show that the value of A produced by Equation (8.9) is correct, we must show that ai = A mod mi for 1 … i … k. Note that cj K Mj K 0 (mod mi) if j ≠ i, and that ci K 1 (mod mi). It follows that ai = A mod mi.

The second assertion of the CRT, concerning arithmetic operations, follows from the rules for modular arithmetic. That is, the second assertion can be stated as follows: If

A 4 (a1, a2, c, ak)

B 4 (b1, b2, c, bk)

then

(A + B) mod M 4 ((a1 + b1) mod m1, c, (ak + bk) mod mk)

(A - B) mod M 4 ((a1 - b1) mod m1, c, (ak - bk) mod mk)

(A * B) mod M 4 ((a1 * b1) mod m1, c, (ak * bk) mod mk)

One of the useful features of the Chinese remainder theorem is that it provides a way to manipulate (potentially very large) numbers mod M in terms of tuples of


smaller numbers. This can be useful when M is 150 digits or more. However, note that it is necessary to know beforehand the factorization of M.

To represent 973 mod 1813 as a pair of numbers mod 37 and 49, define

m1 = 37m2 = 49 M = 1813 A = 973

We also have M1 = 49 and M2 = 37. Using the extended Euclidean algorithm, we compute M1

-1 = 34 mod m1 and M2-1 = 4 mod m2. (Note that we only need

to compute each Mi and each Mi-1 once.) Taking residues modulo 37 and

49, our representation of 973 is (11, 42), because 973 mod 37 = 11 and 973 mod 49 = 42.

Now suppose we want to add 678 to 973. What do we do to (11, 42)? First we compute (678) 4 (678 mod 37, 678 mod 49) = (12, 41). Then we add the tuples element-wise and reduce (11 + 12 mod 37, 42 + 41 mod 49) = (23, 34). To verify that this has the correct effect, we compute

(23, 34) 4 a1M1M1-1 + a2M2M2

-1 mod M

= [(23)(49)(34) + (34)(37)(4)] mod 1813

= 43350 mod 1813

= 1651

and check that it is equal to (973 + 678) mod 1813 = 1651. Remember that in the above derivation, Mi

-1 is the multiplicative inverse of M1 modulo m1 modulo M2

-1 is the multiplicative inverse of M2 modulo m2.Suppose we want to multiply 1651 (mod 1813) by 73. We multiply (23, 34)

by 73 and reduce to get (23 * 73 mod 37, 34 * 73 mod 49) = (14, 32). It is easily verified that

(14, 32) 4 [(14)(49)(34) + (32)(37)(4)] mod 1813

= 865

= 1651 * 73 mod 1813

8.5 discrete LOgarithms

Discrete logarithms are fundamental to a number of public-key algorithms, including Diffie-Hellman key exchange and the digital signature algorithm (DSA). This section provides a brief overview of discrete logarithms. For the interested reader, more detailed developments of this topic can be found in [ORE67] and [LEVE90].

8.5 / disCrete logarithMs 245

The Powers of an Integer, Modulo n

Recall from Euler’s theorem [Equation (8.4)] that, for every a and n that are rela-tively prime,

af(n) K 1 (mod n)

where f(n), Euler’s totient function, is the number of positive integers less than n and relatively prime to n. Now consider the more general expression:

am K 1 (mod n) (8.10)

If a and n are relatively prime, then there is at least one integer m that satisfies Equation (8.10), namely, M = f(n). The least positive exponent m for which Equation (8.10) holds is referred to in several ways:

• The order of a (mod n)

• The exponent to which a belongs (mod n)

• The length of the period generated by a

To see this last point, consider the powers of 7, modulo 19:

71 K 7 (mod 19)72 = 49 = 2 * 19 + 11 K 11 (mod 19)73 = 343 = 18 * 19 + 1 K 1 (mod 19)74 = 2401 = 126 * 19 + 7 K 7 (mod 19)75 = 16807 = 884 * 19 + 11 K 11 (mod 19)

There is no point in continuing because the sequence is repeating. This can be proven by noting that 73 K 1(mod 19), and therefore, 73 + j K 737j K 7j(mod 19), and hence, any two powers of 7 whose exponents differ by 3 (or a multiple of 3) are congruent to each other (mod 19). In other words, the sequence is period-ic, and the length of the period is the smallest positive exponent m such that 7m K 1(mod 19).

Table 8.3 shows all the powers of a, modulo 19 for all positive a 6 19. The length of the sequence for each base value is indicated by shading. Note the following:

1. All sequences end in 1. This is consistent with the reasoning of the preceding few paragraphs.

2. The length of a sequence divides f(19) = 18. That is, an integral number of sequences occur in each row of the table.

3. Some of the sequences are of length 18. In this case, it is said that the base integer a generates (via powers) the set of nonzero integers modulo 19. Each such integer is called a primitive root of the modulus 19.


More generally, we can say that the highest possible exponent to which a num-ber can belong (mod n) is f(n). If a number is of this order, it is referred to as a primitive root of n. The importance of this notion is that if a is a primitive root of n, then its powers

a, a2, c, af(n)

are distinct (mod n) and are all relatively prime to n. In particular, for a prime num-ber p, if a is a primitive root of p, then

a, a2, c, ap - 1

are distinct (mod p). For the prime number 19, its primitive roots are 2, 3, 10, 13, 14, and 15.

Not all integers have primitive roots. In fact, the only integers with primitive roots are those of the form 2, 4, pa, and 2pa, where p is any odd prime and a is a positive integer. The proof is not simple but can be found in many number theory books, including [ORE76].

Logarithms for Modular Arithmetic

With ordinary positive real numbers, the logarithm function is the inverse of expo-nentiation. An analogous function exists for modular arithmetic.

Let us briefly review the properties of ordinary logarithms. The logarithm of a number is defined to be the power to which some positive base (except 1) must be raised in order to equal the number. That is, for base x and for a value y,

y = xlogx(y)

Table 8.3 Powers of Integers, Modulo 19

a a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12 4 8 16 13 7 14 9 18 17 15 11 3 6 12 5 10 13 9 8 5 15 7 2 6 18 16 10 11 14 4 12 17 13 14 16 7 9 17 11 6 5 1 4 16 7 9 17 11 6 5 15 6 11 17 9 7 16 4 1 5 6 11 17 9 7 16 4 16 17 7 4 5 11 9 16 1 6 17 7 4 5 11 9 16 17 11 1 7 11 1 7 11 1 7 11 1 7 11 1 7 11 18 7 18 11 12 1 8 7 18 11 12 1 8 7 18 11 12 19 5 7 6 16 11 4 17 1 9 5 7 6 16 11 4 17 110 5 12 6 3 11 15 17 18 9 14 7 13 16 8 4 2 111 7 1 11 7 1 11 7 1 11 7 1 11 7 1 11 7 112 11 18 7 8 1 12 11 18 7 8 1 12 11 18 7 8 113 17 12 4 14 11 10 16 18 6 2 7 15 5 8 9 3 114 6 8 17 10 7 3 4 18 5 13 11 2 9 12 16 15 115 16 12 9 2 11 13 5 18 4 3 7 10 17 8 6 14 116 9 11 5 4 7 17 6 1 16 9 11 5 4 7 17 6 117 4 11 16 6 7 5 9 1 17 4 11 16 6 7 5 9 118 1 18 1 18 1 18 1 18 1 18 1 18 1 18 1 18 1

8.5 / disCrete logarithMs 247

The properties of logarithms include

logx(1) = 0logx(x) = 1

logx(yz) = logx(y) + logx(z) (8.11)

logx(yr) = r * logx(y) (8.12)

Consider a primitive root a for some prime number p (the argument can be developed for nonprimes as well). Then we know that the powers of a from 1 through (p - 1) produce each integer from 1 through (p - 1) exactly once. We also know that any integer b satisfies

b K r (mod p) for some r, where 0 … r … (p - 1)

by the definition of modular arithmetic. It follows that for any integer b and a primi-tive root a of prime number p, we can find a unique exponent i such that

b K ai(mod p) where 0 … i … (p - 1)

This exponent i is referred to as the discrete logarithm of the number b for the base a (mod p). We denote this value as dloga,p(b).9

Note the following:

dloga,p(1) = 0 because a0 mod p = 1 mod p = 1 (8.13)

dloga,p(a) = 1 because a1 mod p = a (8.14)

9Many texts refer to the discrete logarithm as the index. There is no generally agreed notation for this concept, much less an agreed name.

Here is an example using a nonprime modulus, n = 9. Here f(n) = 6 and a = 2 is a primitive root. We compute the various powers of a and find

20 = 1 24 K 7 (mod 9)21 = 2 25 K 5 (mod 9)22 = 4 26 K 1 (mod 9)23 = 8

This gives us the following table of the numbers with given discrete logarithms (mod 9) for the root a = 2:

Logarithm 0 1 2 3 4 5Number 1 2 4 8 7 5

To make it easy to obtain the discrete logarithms of a given number, we rearrange the table:

Number 1 2 4 5 7 8Logarithm 0 1 2 5 4 3


Now consider

x = adloga, p(x) mod p y = adloga, p(y) mod pxy = adloga, p(xy) mod p

Using the rules of modular multiplication,

xy mod p = [(x mod p)(y mod p)]mod p

adloga, p(xy) mod p = 3 1adloga, p(x) mod p21adloga, p(y) mod p2 4 mod p

= 1adloga, p(x) + dloga, p(y)2 mod p

But now consider Euler’s theorem, which states that, for every a and n that are relatively prime,

af(n) K 1 (mod n)

Any positive integer z can be expressed in the form z = q + kf(n), with 0 … q 6 f(n). Therefore, by Euler’s theorem,

az K aq(mod n) if z K q mod f(n)

Applying this to the foregoing equality, we have

dloga, p(xy) K [dloga, p(x) + dloga, p(y)](modf(p))

and generalizing,

dloga, p(yr) K [r * dloga, p(y)](mod f(p))

This demonstrates the analogy between true logarithms and discrete logarithms.Keep in mind that unique discrete logarithms mod m to some base a exist only

if a is a primitive root of m.Table 8.4, which is directly derived from Table 8.3, shows the sets of discrete

logarithms that can be defined for modulus 19.

Calculation of Discrete Logarithms

Consider the equation

y = gx mod p

Given g, x, and p, it is a straightforward matter to calculate y. At the worst, we must perform x repeated multiplications, and algorithms exist for achieving greater efficiency (see Chapter 9).

However, given y, g, and p, it is, in general, very difficult to calculate x (take the discrete logarithm). The difficulty seems to be on the same order of magnitude as that of factoring primes required for RSA. At the time of this writing, the asymp-totically fastest known algorithm for taking discrete logarithms modulo a prime number is on the order of [BETH91]:

e((ln p)1/3(ln(ln p))2/3))

which is not feasible for large primes.

8.6 / reCoMMeNded readiNg 249


There are many basic texts on the subject of number theory that provide far more detail than most readers of this book will desire. An elementary but nevertheless useful short introduc-tion is [ORE67]. For the reader interested in a more in-depth treatment, two excellent text-books on the subject are [KUMA98] and [ROSE10]. [LEVE90] is a readable and detailed account as well. All of these books include problems with solutions, enhancing their value for self-study.

For readers willing to commit the time, perhaps the best way to get a solid grasp of the fundamentals of number theory is to work their way through [BURN97], which consists solely of a series of exercises with solutions that lead the student step-by-step through the concepts of number theory; working through all of the exercises is equivalent to completing an undergraduate course in number theory.

Table 8.4 Tables of Discrete Logarithms, Modulo 19

(a) Discrete logarithms to the base 2, modulo 19

a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

log2,19(a) 18 1 13 2 16 14 6 3 8 17 12 15 5 7 11 4 10 9

(b) Discrete logarithms to the base 3, modulo 19

a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

log3,19(a) 18 7 1 14 4 8 6 3 2 11 12 15 17 13 5 10 16 9

(c) Discrete logarithms to the base 10, modulo 19

a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

log10,19(a) 18 17 5 16 2 4 12 15 10 1 6 3 13 11 7 14 8 9

(d) Discrete logarithms to the base 13, modulo 19

a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

log13,19(a) 18 11 17 4 14 10 12 15 16 7 6 3 1 5 13 8 2 9

(e) Discrete logarithms to the base 14, modulo 19

a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

log14,19(a) 18 13 7 8 10 2 6 3 14 5 12 15 11 1 17 16 4 9

(f) Discrete logarithms to the base 15, modulo 19

a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

log15,19(a) 18 5 11 10 8 16 12 15 4 13 6 3 7 17 1 2 14 9


8.7 Key terms, review QuestiOns, and PrObLems

Key Terms

bijectioncomposite numberChinese remainder theoremdiscrete logarithm

Euler’s theoremEuler’s totient functionFermat’s theoremindex

orderprime numberprimitive root

BURN97 Burn, R. A Pathway to Number Theory. Cambridge, England: Cambridge University Press, 1997.

KUMA98 Kumanduri, R., and Romero, C. Number Theory with Computer Applications. Upper Saddle River, NJ: Prentice Hall, 1998.

LEVE90 Leveque, W. Elementary Theory of Numbers. New York: Dover, 1990.ORE67 Ore, O. Invitation to Number Theory. Washington, D.C.: The Mathematical

Association of America, 1967.ROSE10 Rosen, K. Elementary Number Theory and its Applications. Reading, MA:

Addison-Wesley, 2010.

Review Questions 8.1 What is a prime number? 8.2 What is the meaning of the expression a divides b? 8.3 What is Euler’s totient function? 8.4 The Miller-Rabin test can determine if a number is not prime but cannot determine if

a number is prime. How can such an algorithm be used to test for primality? 8.5 What is a primitive root of a number? 8.6 What is the difference between an index and a discrete logarithm?

Problems 8.1 The purpose of this problem is to determine how many prime numbers there

are. Suppose there are a total of n prime numbers, and we list these in order: p1 = 2 6 p2 = 3 6 p3 = 5 6 c 6 pn.

a. Define X = 1 + p1 p2 cpn. That is, X is equal to one plus the product of all the primes. Can we find a prime number Pm that divides X?

b. What can you say about m? c. Deduce that the total number of primes cannot be finite. d. Show that Pn + 1 … 1 + p1 p2 c pn. 8.2 The purpose of this problem is to demonstrate that the probability that two random

numbers are relatively prime is about 0.6. a. Let P = Pr[gcd(a, b) = 1]. Show that P = Pr[gcd(a, b) = d] = P/d2. Hint:

Consider the quantity gcd aad

, bdb .

8.7 / Key terMs, review QuestioNs, aNd probleMs 251

b. The sum of the result of part (a) over all possible values of d is 1. That is gd Ú1Pr[gcd(a, b) = d] = 1. Use this equality to determine the value of P. Hint:

Use the identity a∞

i = 1 1

i2 =p2

6.

8.3 Why is gcd(n, n + 1) = 1 for two consecutive integers n and n + 1? 8.4 Using Fermat’s theorem, find 3201 mod 11. 8.5 Use Fermat’s theorem to find a number a between 0 and 72 with a congruent to 9794

modulo 73. 8.6 Use Fermat’s theorem to find a number x between 0 and 28 with x85 congruent to 6

modulo 29. (You should not need to use any brute-force searching.) 8.7 Use Euler’s theorem to find a number a between 0 and 9 such that a is congruent

to 71000 modulo 10. (Note: This is the same as the last digit of the decimal expansion of 71000.)

8.8 Use Euler’s theorem to find a number x between 0 and 28 with x85 congruent to 6 modulo 35. (You should not need to use any brute-force searching.)

8.9 Notice in Table 8.2 that f(n) is even for n 7 2. This is true for all n 7 2. Give a con-cise argument why this is so.

8.10 Prove the following: If p is prime, then f(pi) = pi - pi- 1. Hint: What numbers have a factor in common with pi?

8.11 It can be shown (see any book on number theory) that if gcd(m, n) = 1 then f(mn) = f(m)f(n). Using this property, the property developed in the preceding problem, and the property that f(p) = p - 1 for p prime, it is straightforward to determine the value of f(n) for any n. Determine the following:

a. f(41) b. f(27) c. f(231) d. f(440) 8.12 It can also be shown that for arbitrary positive integer a, f(a) is given by

f(a) = qt

i = 1[pi

ai - 1(pi - 1)]

where a is given by Equation (8.1), namely: a = P1a1 P2

a2 cPtat. Demonstrate this

result. 8.13 Consider the function: f(n) = number of elements in the set

{a: 0 … a 6 n and gcd(a, n) = 1}. What is this function? 8.14 Although ancient Chinese mathematicians did good work coming up with their re-

mainder theorem, they did not always get it right. They had a test for primality. The test said that n is prime if and only if n divides (2n - 2).

a. Give an example that satisfies the condition using an odd prime. b. The condition is obviously true for n = 2. Prove that the condition is true if n is an

odd prime (proving the if condition) c. Give an example of an odd n that is not prime and that does not satisfy the condi-

tion. You can do this with nonprime numbers up to a very large value. This misled the Chinese mathematicians into thinking that if the condition is true then n is prime.

d. Unfortunately, the ancient Chinese never tried n = 341, which is nonprime (341 = 11 * 31), yet 341 divides 2341 - 2 without remainder. Demonstrate that 2341 K 2 (mod 341) (disproving the only if condition). Hint: It is not necessary to calculate 2341; play around with the congruences instead.

8.15 Show that, if n is an odd composite integer, then the Miller-Rabin test will return inconclusive for a = 1 and a = (n - 1).

8.16 If n is composite and passes the Miller-Rabin test for the base a, then n is called a strong pseudoprime to the base a. Show that 2047 is a strong pseudoprime to the base 2.


8.17 A common formulation of the Chinese remainder theorem (CRT) is as follows: Let m1, c, mk be integers that are pairwise relatively prime for 1 … i, j … k, and i ≠ j. Define M to be the product of all the mi′s. Let a1, c, ak be integers. Then the set of congruences:

x K a1(mod m1)

x K a2(mod m2)###

x K ak(mod mk)

has a unique solution modulo M. Show that the theorem stated in this form is true. 8.18 The example used by Sun-Tsu to illustrate the CRT was

x K 2 (mod 3); x K 3 (mod 5); x K 2 (mod 7)

Solve for x. 8.19 Six professors begin courses on Monday, Tuesday, Wednesday, Thursday, Friday,

and Saturday, respectively, and announce their intentions of lecturing at intervals of 2, 3, 4, 1, 6, and 5 days, respectively. The regulations of the university forbid Sunday lectures (so that a Sunday lecture must be omitted). When first will all six professors find themselves compelled to omit a lecture? Hint: Use the CRT.

8.20 Find all primitive roots of 25. 8.21 Given 2 as a primitive root of 29, construct a table of discrete logarithms, and use it to

solve the following congruences. a. 17x2 K 10 (mod 29) b. x2 - 4x - 16 K 0 (mod 29) c. x7 K 17 (mod 29)

Programming Problems 8.22 Write a computer program that implements fast exponentiation (successive squaring)

modulo n. 8.23 Write a computer program that implements the Miller-Rabin algorithm for a user-

specified n. The program should allow the user two choices: (1) specify a possible witness a to test using the Witness procedure or (2) specify a number s of random witnesses for the Miller-Rabin test to check.

9.1 Principles of Public-Key Cryptosystems

Public-Key Cryptosystems Applications for Public-Key Cryptosystems Requirements for Public-Key Cryptography Public-Key Cryptanalysis

9.2 The Rsa algorithm

Description of the Algorithm Computational Aspects The Security of RSA



appendix 9a The Complexity of algorithms

Chapter

Public-Key cryPtograPhy and rSa

253

254 Chapter 9 / publiC-Key Cryptography and rSa

Every Egyptian received two names, which were known respectively as the true name and the good name, or the great name and the little name; and while the good or little name was made public, the true or great name appears to have been care-fully concealed.


Learning Objectives


u Present an overview of the basic principles of public-key cryptosystems.

u Explain the two distinct uses of public-key cryptosystems.

u List and explain the requirements for a public-key cryptosystem.

u Present an overview of the RSA algorithm.

u Understand the timing attack.

u Summarize the relevant issues related to the complexity of algorithms.

The development of public-key cryptography is the greatest and perhaps the only true revolution in the entire history of cryptography. From its earliest begin-nings to modern times, virtually all cryptographic systems have been based on the elementary tools of substitution and permutation. After millennia of working with algorithms that could be calculated by hand, a major advance in symmetric cryptography occurred with the development of the rotor encryption/decryption machine. The electromechanical rotor enabled the development of fiendishly com-plex cipher systems. With the availability of computers, even more complex systems were devised, the most prominent of which was the Lucifer effort at IBM that culmi-nated in the Data Encryption Standard (DES). But both rotor machines and DES, although representing significant advances, still relied on the bread-and-butter tools of substitution and permutation.

Public-key cryptography provides a radical departure from all that has gone before. For one thing, public-key algorithms are based on mathematical functions rather than on substitution and permutation. More important, public-key cryptog-raphy is asymmetric, involving the use of two separate keys, in contrast to sym-metric encryption, which uses only one key. The use of two keys has profound consequences in the areas of confidentiality, key distribution, and authentication, as we shall see.

Before proceeding, we should mention several common misconceptions con-cerning public-key encryption. One such misconception is that public-key encryp-tion is more secure from cryptanalysis than is symmetric encryption. In fact, the security of any encryption scheme depends on the length of the key and the com-putational work involved in breaking a cipher. There is nothing in principle about

publiC-Key Cryptography and rSa 255

either symmetric or public-key encryption that makes one superior to another from the point of view of resisting cryptanalysis.

A second misconception is that public-key encryption is a general-purpose technique that has made symmetric encryption obsolete. On the contrary, because of the computational overhead of current public-key encryption schemes, there seems no foreseeable likelihood that symmetric encryption will be abandoned. As one of the inventors of public-key encryption has put it [DIFF88], “the restriction of public-key cryptography to key management and signature applications is almost universally accepted.”

Finally, there is a feeling that key distribution is trivial when using public-key encryption, compared to the rather cumbersome handshaking involved with key distribution centers for symmetric encryption. In fact, some form of protocol is needed, generally involving a central agent, and the procedures involved are not simpler nor any more efficient than those required for symmetric encryption (e.g., see analysis in [NEED78]).

This chapter and the next provide an overview of public-key cryptography. First, we look at its conceptual framework. Interestingly, the concept for this technique was developed and published before it was shown to be practical to adopt it. Next, we examine the RSA algorithm, which is the most important en-cryption/decryption algorithm that has been shown to be feasible for public-key encryption. Other important public-key cryptographic algorithms are covered in Chapter 10.

Much of the theory of public-key cryptosystems is based on number theory. If one is prepared to accept the results given in this chapter, an understanding of number theory is not strictly necessary. However, to gain a full appreciation of public-key algorithms, some understanding of number theory is required. Chapter 8 provides the necessary background in number theory.

Table 9.1 defines some key terms.

Table 9.1 Terminology Related to Asymmetric Encryption

asymmetric KeysTwo related keys, a public key and a private key, that are used to perform complementary operations, such as encryption and decryption or signature generation and signature verification.

Public Key CertificateA digital document issued and digitally signed by the private key of a Certification Authority that binds the name of a subscriber to a public key. The certificate indicates that the subscriber identified in the certificate has sole control and access to the corresponding private key.

Public Key (asymmetric) Cryptographic algorithmA cryptographic algorithm that uses two related keys, a public key and a private key. The two keys have the property that deriving the private key from the public key is computationally infeasible.

Public Key Infrastructure (PKI)A set of policies, processes, server platforms, software and workstations used for the purpose of administer-ing certificates and public-private key pairs, including the ability to issue, maintain, and revoke public key certificates.

Source: Glossary of Key Information Security Terms, NIST IR 7298 [KISS06].


9.1 PrinciPLes Of PubLic-Key cryPtOsystems

The concept of public-key cryptography evolved from an attempt to attack two of the most difficult problems associated with symmetric encryption. The first problem is that of key distribution, which is examined in some detail in Chapter 14.

As Chapter 14 discusses, key distribution under symmetric encryption requires either (1) that two communicants already share a key, which somehow has been dis-tributed to them; or (2) the use of a key distribution center. Whitfield Diffie, one of the discoverers of public-key encryption (along with Martin Hellman, both at Stanford University at the time), reasoned that this second requirement negated the very essence of cryptography: the ability to maintain total secrecy over your own communication. As Diffie put it [DIFF88], “what good would it do after all to develop impenetrable cryptosystems, if their users were forced to share their keys with a KDC that could be compromised by either burglary or subpoena?”

The second problem that Diffie pondered, and one that was apparently unrelated to the first, was that of digital signatures. If the use of cryptography was to become widespread, not just in military situations but for commercial and private purposes, then electronic messages and documents would need the equiv-alent of signatures used in paper documents. That is, could a method be devised that would stipulate, to the satisfaction of all parties, that a digital message had been sent by a particular person? This is a somewhat broader requirement than that of authentication, and its characteristics and ramifications are explored in Chapter 13.

Diffie and Hellman achieved an astounding breakthrough in 1976 [DIFF76 a, b] by coming up with a method that addressed both problems and was radically different from all previous approaches to cryptography, going back over four millennia.1

In the next subsection, we look at the overall framework for public-key cryptography. Then we examine the requirements for the encryption/decryption algorithm that is at the heart of the scheme.

Public-Key Cryptosystems

Asymmetric algorithms rely on one key for encryption and a different but related key for decryption. These algorithms have the following important characteristic.

• It is computationally infeasible to determine the decryption key given only knowledge of the cryptographic algorithm and the encryption key.

1Diffie and Hellman first publicly introduced the concepts of public-key cryptography in 1976. Hellman credits Merkle with independently discovering the concept at that same time, although Merkle did not publish until 1978 [MERK78]. In fact, the first unclassified document describing public-key distribution and public-key cryptography was a 1974 project proposal by Merkle (http://merkle.com/1974). However, this is not the true beginning. Admiral Bobby Inman, while director of the National Security Agency (NSA), claimed that public-key cryptography had been discovered at NSA in the mid-1960s [SIMM93]. The first documented introduction of these concepts came in 1970, from the Communications-Electronics Security Group, Britain’s counterpart to NSA, in a classified report by James Ellis [ELLI70]. Ellis referred to the technique as nonsecret encryption and describes the discovery in [ELLI99].

http://merkle.com/1974

9.1 / prinCipleS of publiC-Key CryptoSyStemS 257

In addition, some algorithms, such as RSA, also exhibit the following characteristic.

• Either of the two related keys can be used for encryption, with the other used for decryption.

A public-key encryption scheme has six ingredients (Figure 9.1a; compare with Figure 2.1).

• Plaintext: This is the readable message or data that is fed into the algorithm as input.

Plaintextinput

Bobs'spublic-key

ring


PlaintextoutputEncryption algorithm

(e.g., RSA)Decryption algorithm

Joy

Mike

Mike Bob

Ted

Alice

Alice's publickey

Alice's privatekey

(a) Encryption with public key

Plaintextinput


PlaintextoutputEncryption algorithm

(e.g., RSA)Decryption algorithm

Bob's privatekey

Bob

Bob's publickey

Alice'spublic key

ring

JoyTed

(b) Encryption with private key

X

X

PUa

PUb

PRa

PRb

Y = E[PUa, X]

Y = E[PRb, X]

X =D[PRa, Y]

X =D[PUb, Y]

Alice

Bob Alice

Figure 9.1 Public-Key Cryptography


• Encryption algorithm: The encryption algorithm performs various transfor-mations on the plaintext.

• Public and private keys: This is a pair of keys that have been selected so that if one is used for encryption, the other is used for decryption. The exact trans-formations performed by the algorithm depend on the public or private key that is provided as input.

• Ciphertext: This is the scrambled message produced as output. It depends on the plaintext and the key. For a given message, two different keys will produce two different ciphertexts.

• Decryption algorithm: This algorithm accepts the ciphertext and the matching key and produces the original plaintext.

The essential steps are the following.

1. Each user generates a pair of keys to be used for the encryption and decryp-tion of messages.

2. Each user places one of the two keys in a public register or other accessible file. This is the public key. The companion key is kept private. As Figure 9.1a suggests, each user maintains a collection of public keys obtained from others.

3. If Bob wishes to send a confidential message to Alice, Bob encrypts the mes-sage using Alice’s public key.

4. When Alice receives the message, she decrypts it using her private key. No other recipient can decrypt the message because only Alice knows Alice’s pri-vate key.

With this approach, all participants have access to public keys, and pri-vate keys are generated locally by each participant and therefore need never be distributed. As long as a user’s private key remains protected and secret, incom-ing communication is secure. At any time, a system can change its private key and publish the companion public key to replace its old public key.

Table 9.2 summarizes some of the important aspects of symmetric and public-key encryption. To discriminate between the two, we refer to the key used in sym-metric encryption as a secret key. The two keys used for asymmetric encryption are referred to as the public key and the private key.2 Invariably, the private key is kept secret, but it is referred to as a private key rather than a secret key to avoid confu-sion with symmetric encryption.

Let us take a closer look at the essential elements of a public-key encryption scheme, using Figure 9.2 (compare with Figure 2.2). There is some source A that produces a message in plaintext, X = [X1, X2, … , XM]. The M elements of X are let-ters in some finite alphabet. The message is intended for destination B. B generates

2The following notation is used consistently throughout. A secret key is represented by Km, where m is some modifier; for example, Ka is a secret key owned by user A. A public key is represented by PUa, for user A, and the corresponding private key is PRa. Encryption of plaintext X can be performed with a secret key, a public key, or a private key, denoted by E(Ka, X), E(PUa, X), and E(PRa, X), respectively. Similarly, decryption of ciphertext C can be performed with a secret key, a public key, or a private key, denoted by D(Ka, X), D(PUa, X), and D(PRa, X), respectively.


a related pair of keys: a public key, PUb, and a private key, PRb. PRb is known only to B, whereas PUb is publicly available and therefore accessible by A.

With the message X and the encryption key PUb as input, A forms the ciphertext Y = [Y1, Y2, … , YN]:

Y = E(PUb, X)

The intended receiver, in possession of the matching private key, is able to invert the transformation:

X = D(PRb,Y)

Table 9.2 Conventional and Public-Key Encryption

Conventional Encryption Public-Key Encryption

Needed to Work: Needed to Work:

1. The same algorithm with the same key is used for encryption and decryption.

2. The sender and receiver must share the algorithm and the key.

Needed for Security:

1. The key must be kept secret.

2. It must be impossible or at least impractical to decipher a message if the key is kept secret.

3. Knowledge of the algorithm plus samples of ciphertext must be insufficient to determine the key.

1. One algorithm is used for encryption and a related algorithm for decryption with a pair of keys, one for encryption and one for decryption.

2. The sender and receiver must each have one of the matched pair of keys (not the same one).

Needed for Security:

1. One of the two keys must be kept secret.

2. It must be impossible or at least impractical to decipher a message if one of the keys is kept secret.

3. Knowledge of the algorithm plus one of the keys plus samples of ciphertext must be insufficient to determine the other key.

Messagesource

Cryptanalyst

Key pairsource

DestinationX

^PRb

PUb

Encryptionalgorithm

Decryptionalgorithm

PRb

^X

Source A Destination B

Y = E[PUb, X] X =D[PRb, Y]

Figure 9.2 Public-Key Cryptosystem: Secrecy


An adversary, observing Y and having access to PUb, but not having access to PRb or X, must attempt to recover X and/or PRb. It is assumed that the adversary does have knowledge of the encryption (E) and decryption (D) algorithms. If the ad-versary is interested only in this particular message, then the focus of effort is to recover X by generating a plaintext estimate Xn . Often, however, the adversary is interested in being able to read future messages as well, in which case an attempt is made to recover PRb by generating an estimate PRnb.

We mentioned earlier that either of the two related keys can be used for en-cryption, with the other being used for decryption. This enables a rather differ-ent cryptographic scheme to be implemented. Whereas the scheme illustrated in Figure 9.2 provides confidentiality, Figures 9.1b and 9.3 show the use of public-key encryption to provide authentication:

Y = E(PRa, X)

X = D(PUa,Y)

In this case, A prepares a message to B and encrypts it using A’s private key before transmitting it. B can decrypt the message using A’s public key. Because the message was encrypted using A’s private key, only A could have prepared the message. Therefore, the entire encrypted message serves as a digital signature. In addition, it is impossible to alter the message without access to A’s private key, so the message is authenticated both in terms of source and in terms of data integrity.

In the preceding scheme, the entire message is encrypted, which, although val-idating both author and contents, requires a great deal of storage. Each document

Messagesource

Cryptanalyst

Key pairsource

DestinationX

^

PRa

PRa

PUa

Encryptionalgorithm

Decryptionalgorithm


Y = E[PRa, X] X =D[PUa, Y]

Figure 9.3 Public-Key Cryptosystem: Authentication


must be kept in plaintext to be used for practical purposes. A copy also must be stored in ciphertext so that the origin and contents can be verified in case of a dis-pute. A more efficient way of achieving the same results is to encrypt a small block of bits that is a function of the document. Such a block, called an authenticator, must have the property that it is infeasible to change the document without chang-ing the authenticator. If the authenticator is encrypted with the sender’s private key, it serves as a signature that verifies origin, content, and sequencing. Chapter 13 examines this technique in detail.

It is important to emphasize that the encryption process depicted in Figures 9.1b and 9.3 does not provide confidentiality. That is, the message being sent is safe from alteration but not from eavesdropping. This is obvious in the case of a signature based on a portion of the message, because the rest of the message is transmitted in the clear. Even in the case of complete encryption, as shown in Figure 9.3, there is no protection of confidentiality because any observer can decrypt the message by using the sender’s public key.

It is, however, possible to provide both the authentication function and confi-dentiality by a double use of the public-key scheme (Figure 9.4):

Z = E(PUb, E(PRa, X))

X = D(PUa, D(PRb, Z))

In this case, we begin as before by encrypting a message, using the sender’s private key. This provides the digital signature. Next, we encrypt again, using the receiver’s public key. The final ciphertext can be decrypted only by the intended receiver, who alone has the matching private key. Thus, confidentiality is provided. The disadvan-tage of this approach is that the public-key algorithm, which is complex, must be exercised four times rather than two in each communication.

Messagesource

Messagedest.

X Encryptionalgorithm

Key pairsource

PUb PRb


Key pairsource

PRa PUa

Y Encryptionalgorithm

Z Decryptionalgorithm

Y Decryptionalgorithm

X

Figure 9.4 Public-Key Cryptosystem: Authentication and Secrecy


Applications for Public-Key Cryptosystems

Before proceeding, we need to clarify one aspect of public-key cryptosystems that is otherwise likely to lead to confusion. Public-key systems are characterized by the use of a cryptographic algorithm with two keys, one held private and one available pub-licly. Depending on the application, the sender uses either the sender’s private key or the receiver’s public key, or both, to perform some type of cryptographic function. In broad terms, we can classify the use of public-key cryptosystems into three categories

• Encryption/decryption: The sender encrypts a message with the recipient’s public key.

• Digital signature: The sender “signs” a message with its private key. Signing is achieved by a cryptographic algorithm applied to the message or to a small block of data that is a function of the message.

• Key exchange: Two sides cooperate to exchange a session key. Several differ-ent approaches are possible, involving the private key(s) of one or both parties.

Some algorithms are suitable for all three applications, whereas others can be used only for one or two of these applications. Table 9.3 indicates the applications supported by the algorithms discussed in this book.

Requirements for Public-Key Cryptography

The cryptosystem illustrated in Figures 9.2 through 9.4 depends on a cryptographic algorithm based on two related keys. Diffie and Hellman postulated this system without demonstrating that such algorithms exist. However, they did lay out the conditions that such algorithms must fulfill [DIFF76b].

1. It is computationally easy for a party B to generate a pair (public key PUb, private key PRb).

2. It is computationally easy for a sender A, knowing the public key and the mes-sage to be encrypted, M, to generate the corresponding ciphertext:

C = E(PUb, M)

3. It is computationally easy for the receiver B to decrypt the resulting ciphertext using the private key to recover the original message:

M = D(PRb, C) = D[PRb, E(PUb, M)]

4. It is computationally infeasible for an adversary, knowing the public key, PUb, to determine the private key, PRb.

Table 9.3 Applications for Public-Key Cryptosystems

algorithm Encryption/Decryption Digital signature Key Exchange

RSA Yes Yes Yes

Elliptic Curve Yes Yes Yes

Diffie-Hellman No No Yes

DSS No Yes No


5. It is computationally infeasible for an adversary, knowing the public key, PUb, and a ciphertext, C, to recover the original message, M.

We can add a sixth requirement that, although useful, is not necessary for all public-key applications:

6. The two keys can be applied in either order:

M = D[PUb, E(PRb, M)] = D[PRb, E(PUb, M)]

These are formidable requirements, as evidenced by the fact that only a few algorithms (RSA, elliptic curve cryptography, Diffie-Hellman, DSS) have received widespread acceptance in the several decades since the concept of public-key cryp-tography was proposed.

Before elaborating on why the requirements are so formidable, let us first re-cast them. The requirements boil down to the need for a trap-door one-way func-tion. A one-way function3 is one that maps a domain into a range such that every function value has a unique inverse, with the condition that the calculation of the function is easy, whereas the calculation of the inverse is infeasible:

Y = f(X) easy

X = f - 1(Y) infeasible

Generally, easy is defined to mean a problem that can be solved in polynomial time as a function of input length. Thus, if the length of the input is n bits, then the time to compute the function is proportional to na, where a is a fixed constant. Such algorithms are said to belong to the class P. The term infeasible is a much fuzzier concept. In general, we can say a problem is infeasible if the effort to solve it grows faster than polynomial time as a function of input size. For example, if the length of the input is n bits and the time to compute the function is proportional to 2n, the problem is considered infeasible. Unfortunately, it is difficult to determine if a particular algorithm exhibits this complexity. Furthermore, traditional notions of computational complexity focus on the worst-case or average-case complexity of an algorithm. These measures are inadequate for cryptography, which requires that it be infeasible to invert a function for virtually all inputs, not for the worst case or even average case. A brief introduction to some of these concepts is provided in Appendix 9A.

We now turn to the definition of a trap-door one-way function, which is easy to calculate in one direction and infeasible to calculate in the other direction un-less certain additional information is known. With the additional information the inverse can be calculated in polynomial time. We can summarize as follows: A trap-door one-way function is a family of invertible functions fk, such that

Y = fk(X) easy, if k and X are known

X = fk- 1(Y) easy, if k and Y are known

X = fk- 1(Y) infeasible, if Y is known but k is not known

3Not to be confused with a one-way hash function, which takes an arbitrarily large data field as its argument and maps it to a fixed output. Such functions are used for authentication (see Chapter 11).


Thus, the development of a practical public-key scheme depends on discovery of a suitable trap-door one-way function.

Public-Key Cryptanalysis

As with symmetric encryption, a public-key encryption scheme is vulnerable to a brute-force attack. The countermeasure is the same: Use large keys. However, there is a tradeoff to be considered. Public-key systems depend on the use of some sort of invertible mathematical function. The complexity of calculating these functions may not scale linearly with the number of bits in the key but grow more rapidly than that. Thus, the key size must be large enough to make brute-force attack impractical but small enough for practical encryption and decryption. In practice, the key sizes that have been proposed do make brute-force attack impractical but result in encryption/ decryption speeds that are too slow for general-purpose use. Instead, as was men-tioned earlier, public-key encryption is currently confined to key management and signature applications.

Another form of attack is to find some way to compute the private key given the public key. To date, it has not been mathematically proven that this form of at-tack is infeasible for a particular public-key algorithm. Thus, any given algorithm, including the widely used RSA algorithm, is suspect. The history of cryptanalysis shows that a problem that seems insoluble from one perspective can be found to have a solution if looked at in an entirely different way.

Finally, there is a form of attack that is peculiar to public-key systems. This is, in essence, a probable-message attack. Suppose, for example, that a message were to be sent that consisted solely of a 56-bit DES key. An adversary could encrypt all pos-sible 56-bit DES keys using the public key and could discover the encrypted key by matching the transmitted ciphertext. Thus, no matter how large the key size of the public-key scheme, the attack is reduced to a brute-force attack on a 56-bit key. This attack can be thwarted by appending some random bits to such simple messages.

9.2 the rsa aLgOrithm

The pioneering paper by Diffie and Hellman [DIFF76b] introduced a new approach to cryptography and, in effect, challenged cryptologists to come up with a crypto-graphic algorithm that met the requirements for public-key systems. A number of algorithms have been proposed for public-key cryptography. Some of these, though initially promising, turned out to be breakable.4

One of the first successful responses to the challenge was developed in 1977 by Ron Rivest, Adi Shamir, and Len Adleman at MIT and first published in 1978 [RIVE78].5 The Rivest-Shamir-Adleman (RSA) scheme has since that time reigned supreme as the most widely accepted and implemented general-purpose approach to public-key encryption.

4The most famous of the fallen contenders is the trapdoor knapsack proposed by Ralph Merkle. We describe this in Appendix J.5Apparently, the first workable public-key system for encryption/decryption was put forward by Clifford Cocks of Britain’s CESG in 1973 [COCK73]; Cocks’ method is virtually identical to RSA.

9.2 / the rSa algorithm 265

The Rsa scheme is a cipher in which the plaintext and ciphertext are integers between 0 and n - 1 for some n. A typical size for n is 1024 bits, or 309 decimal digits. That is, n is less than 21024. We examine RSA in this section in some detail, beginning with an explanation of the algorithm. Then we examine some of the com-putational and cryptanalytical implications of RSA.

Description of the Algorithm

RSA makes use of an expression with exponentials. Plaintext is encrypted in blocks, with each block having a binary value less than some number n. That is, the block size must be less than or equal to log2(n) + 1; in practice, the block size is i bits, where 2i 6 n … 2i+1. Encryption and decryption are of the following form, for some plaintext block M and ciphertext block C.

C = Me mod n

M = C d mod n = (Me)d mod n = Med mod n

Both sender and receiver must know the value of n. The sender knows the value of e, and only the receiver knows the value of d. Thus, this is a public-key en-cryption algorithm with a public key of PU = {e, n} and a private key of PR = {d, n}. For this algorithm to be satisfactory for public-key encryption, the following require-ments must be met.

1. It is possible to find values of e, d, and n such that Med mod n = M for all M 6 n.

2. It is relatively easy to calculate Me mod n and Cd mod n for all values of M 6 n.

3. It is infeasible to determine d given e and n.

For now, we focus on the first requirement and consider the other questions later. We need to find a relationship of the form

Med mod n = M

The preceding relationship holds if e and d are multiplicative inverses modulo f(n), where f(n) is the Euler totient function. It is shown in Chapter 8 that for p, q prime, f(pq) = (p - 1)(q - 1). The relationship between e and d can be expressed as

ed mod f(n) = 1 (9.1)

This is equivalent to saying

ed K 1 mod f(n)

d K e - 1 mod f(n)

That is, e and d are multiplicative inverses mod f(n). Note that, according to the rules of modular arithmetic, this is true only if d (and therefore e) is relatively prime to f(n). Equivalently, gcd(f(n), d) = 1. See Appendix R for a proof that Equation (9.1) satisfies the requirement for RSA.

We are now ready to state the RSA scheme. The ingredients are the following:

p, q, two prime numbers (private, chosen)n = pq (public, calculated)e, with gcd(f(n), e) = 1; 1 6 e 6 f(n) (public, chosen)d K e-1 (mod f(n)) (private, calculated)


The private key consists of {d, n} and the public key consists of {e, n}. Suppose that user A has published its public key and that user B wishes to send the message M to A. Then B calculates C = Me mod n and transmits C. On receipt of this cipher-text, user A decrypts by calculating M = Cd mod n.

Figure 9.5 summarizes the RSA algorithm. It corresponds to Figure 9.1a: Alice generates a public/private key pair; Bob encrypts using Alice’s public key; and Alice decrypts using her private key. An example from [SING99] is shown in Figure 9.6. For this example, the keys were generated as follows.

1. Select two prime numbers, p = 17 and q = 11.

2. Calculate n = pq = 17 * 11 = 187.

3. Calculate f(n) = (p - 1)(q - 1) = 16 * 10 = 160.

4. Select e such that e is relatively prime to f(n) = 160 and less than f(n); we choose e = 7.

5. Determine d such that de K 1 (mod 160) and d 6 160. The correct value is d = 23, because 23 * 7 = 161 = (1 * 160) + 1; d can be calculated using the extended Euclid’s algorithm (Chapter 4).

The resulting keys are public key PU = {7, 187} and private key PR = {23, 187}. The example shows the use of these keys for a plaintext input of M = 88. For encryption, we need to calculate C = 887 mod 187. Exploiting the properties of modular arithmetic, we can do this as follows.

887 mod 187 = [(884 mod 187) * (882 mod 187) * (881 mod 187)] mod 187

881 mod 187 = 88

882 mod 187 = 7744 mod 187 = 77

884 mod 187 = 59,969,536 mod 187 = 132

887 mod 187 = (88 * 77 * 132) mod 187 = 894,432 mod 187 = 11

For decryption, we calculate M = 1123 mod 187:

1123 mod 187 = [(111 mod 187) * (112 mod 187) * (114 mod 187) * (118 mod 187) * (118 mod 187)] mod 187

111 mod 187 = 11

112 mod 187 = 121

114 mod 187 = 14,641 mod 187 = 55

118 mod 187 = 214,358,881 mod 187 = 33

1123 mod 187 = (11 * 121 * 55 * 33 * 33) mod 187 = 79,720,245 mod 187 = 88

We now look at an example from [HELL79], which shows the use of RSA to process multiple blocks of data. In this simple example, the plaintext is an alpha-numeric string. Each plaintext symbol is assigned a unique code of two decimal


digits (e.g., a = 00, A = 26).6 A plaintext block consists of four decimal digits, or two alphanumeric characters. Figure 9.7a illustrates the sequence of events for the encryption of multiple blocks, and Figure 9.7b gives a specific example. The circled numbers indicate the order in which operations are performed.

Computational Aspects

We now turn to the issue of the complexity of the computation required to use RSA. There are actually two issues to consider: encryption/decryption and key generation. Let us look first at the process of encryption and decryption and then consider key generation.

Encryption

Plaintext88

Plaintext88

Ciphertext11

88 mod 187 � 11

PU � 7, 187

Decryption

711 mod 187 � 88

PR � 23, 187

23

Figure 9.6 Example of RSA Algorithm

6The complete mapping of alphanumeric characters to decimal digits is at this book’s Premium Content Web site in the document RSAexample.pdf.

Figure 9.5 The RSA Algorithm

Encryption by Bob with alice’s Public Key

Plaintext: M 6 n

Ciphertext: C = Me mod n

Decryption by alice with alice’s Public Key

Ciphertext: C

Plaintext: M = Cd mod n

Key Generation by alice

Select p, q p and q both prime, p ≠ q

Calculate n = p * q

Calcuate f(n) = (p - 1)(q - 1)

Select integer e gcd (f(n), e) = 1; 1 6 e 6 f(n)

Calculate d d K e-1 (mod f(n))

Public key PU = {e, n}

Private key PR = {d, n}


ExponEntiation in Modular arithMEtic Both encryption and decryption in RSA involve raising an integer to an integer power, mod n. If the exponentiation is done over the integers and then reduced modulo n, the intermediate values would be gargantuan. Fortunately, as the preceding example shows, we can make use of a property of modular arithmetic:

[(a mod n) * (b mod n)] mod n = (a * b) mod n

Thus, we can reduce intermediate results modulo n. This makes the calculation practical.

Another consideration is the efficiency of exponentiation, because with RSA, we are dealing with potentially large exponents. To see how efficiency might be in-creased, consider that we wish to compute x16. A straightforward approach requires 15 multiplications:

x16 = x * x * x * x * x * x * x * x * x * x * x * x * x * x * x * x

Plaintext P

Decimal string

Sender

Receiver

(a) General approach (b) Example

Blocks of numbers

Transmit

P1, P2,

P1 = C1d mod n

P2 = C2d mod n

Ciphertext C

C1 = P1e mod n

C2 = P2e mod n

Recovereddecimal text

n = pq

Random numbergenerator

e, p, q

Private keyd, n

Public keye, n

How_are_you?

33 14 22 62 00 17 04 62 24 14 20 66

Sender

Receiver

Transmit

P1 = 3314 P2 = 2262 P3 = 0017P4 = 0462 P5 = 2414 P6 = 2066

C1 = 331411 mod 11023 = 10260C2 = 226211 mod 11023 = 9489C3 = 1711 mod 11023 = 1782C4 = 46211 mod 11023 = 727C5 = 241411 mod 11023 = 10032C6 = 206611 mod 11023 = 2253

P1 = 102605891 mod 11023 = 3314P2 = 94895891 mod 11023 = 2262P3 = 17825891 mod 11023 = 0017P4 = 7275891 mod 11023 = 0462P5 = 100325891 mod 11023 = 2414P6 = 22535891 mod 11023 = 2066

11023 = 73 151

5891 = 11–1 mod 1080010800 = (73 – 1)(151 – 1)11023 = 73 51

Random numbergenerator

e = 11 n = 11023

d = 5891 n = 11023

e = 11 p = 73, q = 151

1

2

6

3

4

5

7

1

2

6

3

4

5

7

d = e–1 mod φ(n) φ(n) = (p – 1)(q – 1)

n = pq

Figure 9.7 RSA Processing of Multiple Blocks


However, we can achieve the same final result with only four multiplications if we repeatedly take the square of each partial result, successively forming (x2, x4, x8, x16). As another example, suppose we wish to calculate x11 mod n for some integers x and n. Observe that x11 = x1+2+8 = (x)(x2)(x8). In this case, we compute x mod n, x2 mod n, x4 mod n, and x8 mod n and then calculate [(x mod n) * (x2 mod n) * (x8 mod n)] mod n.

More generally, suppose we wish to find the value ab mod n with a, b, and m positive integers. If we express b as a binary number bkbk-1 . . . b0, then we have

b = abi ≠ 0

2i

Therefore,

ab = aa g 2i

bi ≠ 0b

= qbi ≠ 0

a(2i)

abmod n = c qbi ≠ 0

a(2i) dmod n = a qbi ≠ 0

c a(2i)mod n d bmod n

We can therefore develop the algorithm7 for computing ab mod n, shown in Figure 9.8. Table 9.4 shows an example of the execution of this algorithm. Note that the variable c is not needed; it is included for explanatory purposes. The final value of c is the value of the exponent.

EfficiEnt opEration using thE public KEy To speed up the operation of the RSA algorithm using the public key, a specific choice of e is usually made. The most common choice is 65537 (216 + 1); two other popular choices are 3 and 17. Each of these choices has only two 1 bits, so the number of multiplications required to per-form exponentiation is minimized.

7The algorithm has a long history; this particular pseudocode expression is from [CORM09].

c 0; f 1

c 2 � cdo

bi � 1

then c c � 1

if

f (f � f) mod n

f (f � a) mod n

for i k downto 0

return f

Note: The integer b is expressed as a binary number bkbk – 1 … b0.

Figure 9.8 Algorithm for Computing ab mod n


However, with a very small public key, such as e = 3, RSA becomes vulnerable to a simple attack. Suppose we have three different RSA users who all use the value e = 3 but have unique values of n, namely (n1, n2, n3). If user A sends the same en-crypted message M to all three users, then the three ciphertexts are C1 = M3 mod n1, C2 = M3 mod n2, and C3 = M3 mod n3. It is likely that n1, n2, and n3 are pairwise relatively prime. Therefore, one can use the Chinese remainder theorem (CRT) to compute M3 mod (n1n2n3). By the rules of the RSA algorithm, M is less than each of the ni; therefore M3 6 n1n2n3. Accordingly, the attacker need only compute the cube root of M3. This attack can be countered by adding a unique pseudorandom bit string as padding to each instance of M to be encrypted. This approach is discussed subsequently.

The reader may have noted that the definition of the RSA algorithm (Figure 9.5) requires that during key generation the user selects a value of e that is relatively prime to f(n). Thus, if a value of e is selected first and the primes p and q are generated, it may turn out that gcd(f(n), e) ≠ 1. In that case, the user must reject the p, q values and generate a new p, q pair.

EfficiEnt opEration using thE privatE KEy We cannot similarly choose a small constant value of d for efficient operation. A small value of d is vulnerable to a brute-force attack and to other forms of cryptanalysis [WIEN90]. However, there is a way to speed up computation using the CRT. We wish to compute the value M = Cd mod n. Let us define the following intermediate results:

Vp = Cd mod p Vq = Cd mod q

Following the CRT using Equation (8.8), define the quantities

Xp = q * (q- 1 mod p) Xq = p * (p- 1 mod q)

The CRT then shows, using Equation (8.9), that

M = (Vp Xp + Vq Xq) mod n

Furthermore, we can simplify the calculation of Vp and Vq using Fermat’s theorem, which states that a p-1 K 1 (mod p) if p and a are relatively prime. Some thought should convince you that the following are valid.

Vp = Cd mod p = Cd mod (p - 1) mod p Vq = Cd mod q = Cd mod (q - 1) mod q

Table 9.4 Result of the Fast Modular Exponentiation Algorithm for ab mod n, where a = 7, b = 560 = 1000110000, and n = 561

i 9 8 7 6 5 4 3 2 1 0

bi 1 0 0 0 1 1 0 0 0 0c 1 2 4 8 17 35 70 140 280 560f 7 49 157 526 160 241 298 166 67 1


The quantities d mod (p - 1) and d mod (q - 1) can be precalculated. The end result is that the calculation is approximately four times as fast as evaluating M = Cd mod n directly [BONE02].

KEy gEnEration Before the application of the public-key cryptosystem, each participant must generate a pair of keys. This involves the following tasks.

• Determining two prime numbers, p and q.

• Selecting either e or d and calculating the other.

First, consider the selection of p and q. Because the value of n = pq will be known to any potential adversary, in order to prevent the discovery of p and q by exhaustive methods, these primes must be chosen from a sufficiently large set (i.e., p and q must be large numbers). On the other hand, the method used for finding large primes must be reasonably efficient.

At present, there are no useful techniques that yield arbitrarily large primes, so some other means of tackling the problem is needed. The procedure that is gen-erally used is to pick at random an odd number of the desired order of magnitude and test whether that number is prime. If not, pick successive random numbers until one is found that tests prime.

A variety of tests for primality have been developed (e.g., see [KNUT98] for a description of a number of such tests). Almost invariably, the tests are probabi-listic. That is, the test will merely determine that a given integer is probably prime. Despite this lack of certainty, these tests can be run in such a way as to make the probability as close to 1.0 as desired. As an example, one of the more efficient and popular algorithms, the Miller-Rabin algorithm, is described in Chapter 8. With this algorithm and most such algorithms, the procedure for testing whether a given integer n is prime is to perform some calculation that involves n and a randomly chosen integer a. If n “fails” the test, then n is not prime. If n “passes” the test, then n may be prime or nonprime. If n passes many such tests with many different randomly chosen values for a, then we can have high confidence that n is, in fact, prime.

In summary, the procedure for picking a prime number is as follows.

1. Pick an odd integer n at random (e.g., using a pseudorandom number generator).

2. Pick an integer a 6 n at random.

3. Perform the probabilistic primality test, such as Miller-Rabin, with a as a parameter. If n fails the test, reject the value n and go to step 1.

4. If n has passed a sufficient number of tests, accept n; otherwise, go to step 2.

This is a somewhat tedious procedure. However, remember that this process is performed relatively infrequently: only when a new pair (PU, PR) is needed.

It is worth noting how many numbers are likely to be rejected before a prime number is found. A result from number theory, known as the prime num-ber theorem, states that the primes near N are spaced on the average one every


ln (N) integers. Thus, on average, one would have to test on the order of ln(N) integers before a prime is found. Actually, because all even integers can be im-mediately rejected, the correct figure is ln(N)/2. For example, if a prime on the order of magnitude of 2200 were sought, then about ln(2200)/2 = 70 trials would be needed to find a prime.

Having determined prime numbers p and q, the process of key generation is completed by selecting a value of e and calculating d or, alternatively, selecting a value of d and calculating e. Assuming the former, then we need to select an e such that gcd(f(n), e) = 1 and then calculate d K e-1 (mod f(n)). Fortunately, there is a single algorithm that will, at the same time, calculate the greatest com-mon divisor of two integers and, if the gcd is 1, determine the inverse of one of the integers modulo the other. The algorithm, referred to as the extended Euclid’s algorithm, is explained in Chapter 4. Thus, the procedure is to generate a series of random numbers, testing each against f(n) until a number relatively prime to f(n) is found. Again, we can ask the question: How many random numbers must we test to find a usable number, that is, a number relatively prime to f(n)? It can be shown easily that the probability that two random numbers are relatively prime is about 0.6; thus, very few tests would be needed to find a suitable integer (see Problem 8.2).

The Security of RSA

Five possible approaches to attacking the RSA algorithm are

• Brute force: This involves trying all possible private keys.

• Mathematical attacks: There are several approaches, all equivalent in effort to factoring the product of two primes.

• Timing attacks: These depend on the running time of the decryption algorithm.

• Hardware fault-based attack: This involves inducing hardware faults in the processor that is generating digital signatures.

• Chosen ciphertext attacks: This type of attack exploits properties of the RSA algorithm.

The defense against the brute-force approach is the same for RSA as for other cryptosystems, namely, to use a large key space. Thus, the larger the number of bits in d, the better. However, because the calculations involved, both in key generation and in encryption/decryption, are complex, the larger the size of the key, the slower the system will run.

In this subsection, we provide an overview of mathematical and timing attacks.

thE factoring problEM We can identify three approaches to attacking RSA mathematically.

1. Factor n into its two prime factors. This enables calculation of f(n) = (p - 1) * (q - 1), which in turn enables determination of d K e- 1 (mod f(n)).

2. Determine f(n) directly, without first determining p and q. Again, this enables determination of d K e- 1 (mod f(n)).

3. Determine d directly, without first determining f(n).


Most discussions of the cryptanalysis of RSA have focused on the task of factor-ing n into its two prime factors. Determining f(n) given n is equivalent to factoring n [RIBE96]. With presently known algorithms, determining d given e and n appears to be at least as time-consuming as the factoring problem [KALI95]. Hence, we can use factoring performance as a benchmark against which to evaluate the security of RSA.

For a large n with large prime factors, factoring is a hard problem, but it is not as hard as it used to be. A striking illustration of this is the following. In 1977, the three inventors of RSA dared Scientific American readers to decode a cipher they printed in Martin Gardner’s “Mathematical Games” column [GARD77]. They offered a $100 reward for the return of a plaintext sentence, an event they predicted might not occur for some 40 quadrillion years. In April of 1994, a group working over the Internet claimed the prize after only eight months of work [LEUT94]. This challenge used a public key size (length of n) of 129 decimal digits, or around 428 bits. In the meantime, just as they had done for DES, RSA Laboratories had issued challenges for the RSA cipher with key sizes of 100, 110, 120, and so on, digits. The latest challenge to be met is the RSA-768 challenge with a key length of 232 decimal digits, or 768 bits. Table 9.5 shows the results to date. Million-instructions-per-second processor running for one year, which is about 3 * 1013 instructions executed. A 1 GHz Pentium is about a 250-MIPS machine.

A striking fact about the progress reflected in Table 9.5 concerns the method used. Until the mid-1990s, factoring attacks were made using an approach known as the quadratic sieve. The attack on RSA-130 used a newer algorithm, the general-ized number field sieve (GNFS), and was able to factor a larger number than RSA-129 at only 20% of the computing effort.

The threat to larger key sizes is twofold: the continuing increase in comput-ing power and the continuing refinement of factoring algorithms. We have seen that the move to a different algorithm resulted in a tremendous speedup. We can expect further refinements in the GNFS, and the use of an even better algo-rithm is also a possibility. In fact, a related algorithm, the special number field

Table 9.5 Progress in RSA Factorization

Number of Decimal Digits Number of Bits Date achieved

100 332 April 1991

110 365 April 1992

120 398 June 1993

129 428 April 1994

130 431 April 1996

140 465 February 1999

155 512 August 1999

160 530 April 2003

174 576 December 2003

200 663 May 2005

193 640 November 2005

232 768 December 2009


sieve (SNFS), can factor numbers with a specialized form considerably faster than the generalized number field sieve. Figure 9.9 compares the performance of the two algorithms. It is reasonable to expect a breakthrough that would en-able a general factoring performance in about the same time as SNFS, or even better [ODLY95]. Thus, we need to be careful in choosing a key size for RSA. The team that produced the 768-bit factorization made the following observa-tion [KLEI10]:

Factoring a 1024-bit RSA modulus would be about a thousand times harder than factoring a 768-bit modulus, and a 768-bit RSA modulus is several thousands times harder to factor than a 512-bit one. Because the first factorization of a 512-bit RSA modulus

1022

MIP

S-ye

ars

need

ed t

o fa

ctor

200018001600140012001000800600

Bits

General number field sieve

Special number field sieve

1020

1018

1016

1014

1012

1010

108

106

104

102

100

Figure 9.9 MIPS-years Needed to Factor


was reported only a decade, it is not unreasonable to expect that 1024-bit RSA moduli can be factored well within the next decade by an academic effort such as ours. Thus, it would be prudent to phase out usage of 1024-bit RSA within the next three to four years.

In addition to specifying the size of n, a number of other constraints have been suggested by researchers. To avoid values of n that may be factored more easily, the algorithm’s inventors suggest the following constraints on p and q.

1. p and q should differ in length by only a few digits. Thus, for a 1024-bit key (309 decimal digits), both p and q should be on the order of magnitude of 1075 to 10100.

2. Both (p - 1) and (q - 1) should contain a large prime factor.

3. gcd(p - 1, q - 1) should be small.

In addition, it has been demonstrated that if e 6 n and d 6 n1/4, then d can be easily determined [WIEN90].

tiMing attacKs If one needed yet another lesson about how difficult it is to assess the security of a cryptographic algorithm, the appearance of timing attacks provides a stunning one. Paul Kocher, a cryptographic consultant, demonstrated that a snooper can determine a private key by keeping track of how long a computer takes to decipher messages [KOCH96, KALI96b]. Timing attacks are applicable not just to RSA, but to other public-key cryptography systems. This attack is alarm-ing for two reasons: It comes from a completely unexpected direction, and it is a ciphertext-only attack.

A timing attack is somewhat analogous to a burglar guessing the combination of a safe by observing how long it takes for someone to turn the dial from number to number. We can explain the attack using the modular exponentiation algorithm of Figure 9.8, but the attack can be adapted to work with any implementation that does not run in fixed time. In this algorithm, modular exponentiation is accom-plished bit by bit, with one modular multiplication performed at each iteration and an additional modular multiplication performed for each 1 bit.

As Kocher points out in his paper, the attack is simplest to understand in an extreme case. Suppose the target system uses a modular multiplication function that is very fast in almost all cases but in a few cases takes much more time than an entire average modular exponentiation. The attack proceeds bit-by-bit starting with the leftmost bit, bk. Suppose that the first j bits are known (to obtain the entire expo-nent, start with j = 0 and repeat the attack until the entire exponent is known). For a given ciphertext, the attacker can complete the first j iterations of the for loop. The operation of the subsequent step depends on the unknown exponent bit. If the bit is set, d d (d * a) mod n will be executed. For a few values of a and d, the modu-lar multiplication will be extremely slow, and the attacker knows which these are. Therefore, if the observed time to execute the decryption algorithm is always slow when this particular iteration is slow with a 1 bit, then this bit is assumed to be 1. If a number of observed execution times for the entire algorithm are fast, then this bit is assumed to be 0.


In practice, modular exponentiation implementations do not have such extreme timing variations, in which the execution time of a single iteration can exceed the mean execution time of the entire algorithm. Nevertheless, there is enough variation to make this attack practical. For details, see [KOCH96].

Although the timing attack is a serious threat, there are simple countermea-sures that can be used, including the following.

• Constant exponentiation time: Ensure that all exponentiations take the same amount of time before returning a result. This is a simple fix but does degrade performance.

• Random delay: Better performance could be achieved by adding a random delay to the exponentiation algorithm to confuse the timing attack. Kocher points out that if defenders don’t add enough noise, attackers could still succeed by collecting additional measurements to compensate for the random delays.

• Blinding: Multiply the ciphertext by a random number before performing exponentiation. This process prevents the attacker from knowing what cipher-text bits are being processed inside the computer and therefore prevents the bit-by-bit analysis essential to the timing attack.

RSA Data Security incorporates a blinding feature into some of its products. The private-key operation M = Cd mod n is implemented as follows.

1. Generate a secret random number r between 0 and n - 1.

2. Compute C′ = C(r e) mod n, where e is the public exponent.

3. Compute M′ = (C′)d mod n with the ordinary RSA implementation.

4. Compute M = M′r-1 mod n. In this equation, r-1 is the multiplicative inverse of r mod n; see Chapter 4 for a discussion of this concept. It can be demon-strated that this is the correct result by observing that red mod n = r mod n.

RSA Data Security reports a 2 to 10% performance penalty for blinding.

fault-basEd attacK Still another unorthodox approach to attacking RSA is re-ported in [PELL10]. The approach is an attack on a processor that is generating RSA digital signatures. The attack induces faults in the signature computation by reducing the power to the processor. The faults cause the software to produce in-valid signatures, which can then be analyzed by the attacker to recover the private key. The authors show how such an analysis can be done and then demonstrate it by extracting a 1024-bit private RSA key in approximately 100 hours, using a com-mercially available microprocessor.

The attack algorithm involves inducing single-bit errors and observing the re-sults. The details are provided in [PELL10], which also references other proposed hardware fault-based attacks against RSA.

This attack, while worthy of consideration, does not appear to be a serious threat to RSA. It requires that the attacker have physical access to the target machine and that the attacker is able to directly control the input power to the processor. Controlling the input power would for most hardware require more than simply controlling the AC power, but would also involve the power supply control hardware on the chip.


chosEn ciphErtExt attacK and optiMal asyMMEtric Encryption padding The basic RSA algorithm is vulnerable to a chosen ciphertext attack (CCA). CCA is defined as an attack in which the adversary chooses a number of ciphertexts and is then given the corresponding plaintexts, decrypted with the target’s private key. Thus, the adversary could select a plaintext, encrypt it with the target’s public key, and then be able to get the plaintext back by having it decrypted with the private key. Clearly, this provides the adversary with no new information. Instead, the ad-versary exploits properties of RSA and selects blocks of data that, when processed using the target’s private key, yield information needed for cryptanalysis.

A simple example of a CCA against RSA takes advantage of the following property of RSA:

E(PU, M1) * E(PU, M2) = E(PU, [M1 * M2]) (9.2)

We can decrypt C = Me mod n using a CCA as follows.

1. Compute X = (C * 2e) mod n.

2. Submit X as a chosen ciphertext and receive back Y = Xd mod n.

But now note that

X = (C mod n) * (2e mod n) = (Me mod n) * (2e mod n) = (2M)e mod n

Therefore, Y = (2M) mod n. From this, we can deduce M. To overcome this simple attack, practical RSA-based cryptosystems randomly pad the plaintext prior to encryption. This randomizes the ciphertext so that Equation (9.2) no longer holds. However, more sophisticated CCAs are possible, and a simple padding with a random value has been shown to be insufficient to provide the desired security. To counter such attacks, RSA Security Inc., a leading RSA vendor and former holder of the RSA patent, recommends modifying the plaintext using a procedure known as optimal asymmetric encryption padding (OAEP). A full discussion of the threats and OAEP are beyond our scope; see [POIN02] for an introduction and [BELL94] for a thorough analysis. Here, we simply summarize the OAEP procedure.

Figure 9.10 depicts OAEP encryption. As a first step, the message M to be encrypted is padded. A set of optional parameters, P, is passed through a hash function, H.8 The output is then padded with zeros to get the desired length in the overall data block (DB). Next, a random seed is generated and passed through another hash function, called the mask generating function (MGF). The resulting hash value is bit-by-bit XORed with DB to produce a maskedDB. The maskedDB is in turn passed through the MGF to form a hash that is XORed with the seed to produce the maskedseed. The concatenation of the maskedseed and the maskedDB forms the encoded message EM. Note that the EM includes the padded message, masked by the seed, and the seed, masked by the maskedDB. The EM is then encrypted using RSA.

8A hash function maps a variable-length data block or message into a fixed-length value called a hash code. Hash functions are discussed in depth in Chapter 11.



The recommended treatments of encryption listed in Chapter 3 cover public-key as well as symmetric encryption.

[DIFF88] describes in detail the several attempts to devise secure two-key crypto-algorithms and the gradual evolution of a variety of protocols based on them. [CORM09] provides a concise but complete and readable summary of all of the algorithms relevant to the verification, computation, and cryptanalysis of RSA. [BONE99] and [SHAM03] discuss various cryptanalytic attacks on RSA.

Seed

Maskedseed

DB

MaskedDB

M

EM

Padding

H(P)

MGF

MGF

P

P � encoding parametersM � message to be encodedH � hash function

DB � data blockMGF � mask generating functionEM � encoded message

Figure 9.10 Encryption Using Optimal Asymmetric Encryption Padding (OAEP)

BONE99 Boneh, D. “Twenty Years of Attacks on the RSA Cryptosystem.” Notices of the American Mathematical Society, February 1999.

CORM09 Cormen, T.; Leiserson, C.; Rivest, R.; and Stein, C. Introduction to Algorithms. Cambridge, MA: MIT Press, 2009.



Key Terms

chosen ciphertext attack (CCA)digital signaturekey exchangeone-way functionoptimal asymmetric encryption

padding (OAEP)

private keypublic keypublic-key cryptographypublic-key cryptosystemspublic-key encryptionRSA

time complexitytiming attacktrap-door one-way function

Review Questions 9.1 What are the principal elements of a public-key cryptosystem? 9.2 What are the roles of the public and private key? 9.3 What are three broad categories of applications of public-key cryptosystems? 9.4 What requirements must a public-key cryptosystems fulfill to be a secure algorithm? 9.5 What is a one-way function? 9.6 What is a trap-door one-way function? 9.7 Describe in general terms an efficient procedure for picking a prime number.

Problems 9.1 Prior to the discovery of any specific public-key schemes, such as RSA, an existence

proof was developed whose purpose was to demonstrate that public-key encryption is possible in theory. Consider the functions f1(x1) = z1; f2(x2, y2) = z2; f3(x3, y3) = z3, where all values are integers with 1 … xi, yi, zi … N. Function f1 can be represented by a vector M1 of length N, in which the kth entry is the value of f1(k). Similarly, f2 and f3 can be represented by N * N matrices M2 and M3. The intent is to represent the encryption/decryption process by table lookups for tables with very large values of N. Such tables would be impractically huge but could be constructed in principle. The scheme works as follows: Construct M1 with a random permutation of all integers between 1 and N; that is, each integer appears exactly once in M1. Construct M2 so that each row contains a random permutation of the first N integers. Finally, fill in M3 to satisfy the following condition:

f3(f2(f1(k), p), k) = p for all k, p with 1 … k, p … N

To summarize, 1. M1 takes an input k and produces an output x. 2. M2 takes inputs x and p giving output z. 3. M3 takes inputs z and k and produces p.

The three tables, once constructed, are made public.

DIFF88 Diffie, W. “The First Ten Years of Public-Key Cryptography.” Proceedings of the IEEE, May 1988.

sHaM03 Shamir, A., and Tromer, E. “On the Cost of Factoring RSA-1024.” CryptoBytes, Summer 2003. http://www.rsasecurity.com/rsalabs

http://www.rsasecurity.com/rsalabs


a. It should be clear that it is possible to construct M3 to satisfy the preceding condi-tion. As an example, fill in M3 for the following simple case:

M1 =

5

4

2

3

1

M2 =

5 2 3 4 1

4 2 5 1 3

1 3 2 4 5

3 1 4 2 5

2 5 3 4 1

M3 =

Convention: The ith element of M1 corresponds to k = i. The ith row of M2 cor-responds to x = i; the jth column of M2 corresponds to p = j. The ith row of M3 corresponds to z = i; the jth column of M3 corresponds to k = j.

b. Describe the use of this set of tables to perform encryption and decryption between two users.

c. Argue that this is a secure scheme. 9.2 Perform encryption and decryption using the RSA algorithm, as in Figure 9.5, for the

following: a. p = 3; q = 11, e = 7; M = 5 b. p = 5; q = 11, e = 3; M = 9 c. p = 7; q = 11, e = 17; M = 8 d. p = 11; q = 13, e = 11; M = 7 e. p = 17; q = 31, e = 7; M = 2

Hint: Decryption is not as hard as you think; use some finesse. 9.3 In a public-key system using RSA, you intercept the ciphertext C = 10 sent to a user

whose public key is e = 5, n = 35. What is the plaintext M? 9.4 In an RSA system, the public key of a given user is e = 31, n = 3599. What is the pri-

vate key of this user? Hint: First use trial-and-error to determine p and q; then use the extended Euclidean algorithm to find the multiplicative inverse of 31 modulo f(n).

9.5 In using the RSA algorithm, if a small number of repeated encodings give back the plaintext, what is the likely cause?

9.6 Suppose we have a set of blocks encoded with the RSA algorithm and we don’t have the private key. Assume n = pq, e is the public key. Suppose also someone tells us they know one of the plaintext blocks has a common factor with n. Does this help us in any way?

9.7 In the RSA public-key encryption scheme, each user has a public key, e, and a private key, d. Suppose Bob leaks his private key. Rather than generating a new modulus, he decides to generate a new public and a new private key. Is this safe?

9.8 Suppose Bob uses the RSA cryptosystem with a very large modulus n for which the factorization cannot be found in a reasonable amount of time. Suppose Alice sends a message to Bob by representing each alphabetic character as an integer between 0 and 25 (A S 0, c, Z S 25) and then encrypting each number separately using RSA with large e and large n. Is this method secure? If not, describe the most efficient at-tack against this encryption method.

9.9 Using a spreadsheet (such as Excel) or a calculator, perform the operations described below. Document results of all intermediate modular multiplications. Determine a number of modular multiplications per each major transformation (such as encryp-tion, decryption, primality testing, etc.).

a. Test all odd numbers in the range from 233 to 241 for primality using the Miller-Rabin test with base 2.

b. Encrypt the message block M = 2 using RSA with the following parameters: e = 23 and n = 233 * 241.

c. Compute a private key (d, p, q) corresponding to the given above public key (e, n).


d. Perform the decryption of the obtained ciphertext 1. without using the Chinese Remainder Theorem, and 2. using the Chinese Remainder Theorem. 9.10 Assume that you generate an authenticated and encrypted message by first applying

the RSA transformation determined by your private key, and then enciphering the message using recipient’s public key (note that you do NOT use hash function before the first transformation). Will this scheme work correctly [i.e., give the possibility to re-construct the original message at the recipient’s side, for all possible relations between the sender’s modulus nS and the recipient’s modulus nR (nS 7 nR, nS 6 nR, nS = nR)]? Explain your answer. In case your answer is “no,” how would you correct this scheme?

9.11 “I want to tell you, Holmes,” Dr. Watson’s voice was enthusiastic, “that your recent activities in network security have increased my interest in cryptography. And just yesterday I found a way to make one-time pad encryption practical.”

“Oh, really?” Holmes’ face lost its sleepy look.“Yes, Holmes. The idea is quite simple. For a given one-way function F, I gener-

ate a long pseudorandom sequence of elements by applying F to some standard se-quence of arguments. The cryptanalyst is assumed to know F and the general nature of the sequence, which may be as simple as S, S + 1, S + 2, … , but not secret S. And due to the one-way nature of F, no one is able to extract S given F(S + i) for some i, thus even if he somehow obtains a certain segment of the sequence, he will not be able to determine the rest.”

“I am afraid, Watson, that your proposal isn’t without flaws and at least it needs some additional conditions to be satisfied by F. Let’s consider, for instance, the RSA encryption function, that is F(M) = MK mod N, K is secret. This function is believed to be one-way, but I wouldn’t recommend its use, for example, on the sequence M = 2, 3, 4, 5, 6, …”

“But why, Holmes?” Dr. Watson apparently didn’t understand. “Why do you think that the resulting sequence 2K mod N, 3K mod N, 4K mod N, … is not appropri-ate for one-time pad encryption if K is kept secret?”

“Because it is—at least partially—predictable, dear Watson, even if K is kept secret. You have said that the cryptanalyst is assumed to know F and the general nature of the sequence. Now let’s assume that he will obtain somehow a short segment of the output sequence. In crypto circles, this assumption is generally considered to be a viable one. And for this output sequence, knowledge of just the first two elements will allow him to predict quite a lot of the next elements of the sequence, even if not all of them, thus this sequence can’t be considered to be cryptographically strong. And with the knowledge of a longer segment he could predict even more of the next elements of the sequence. Look, knowing the general nature of the sequence and its first two elements 2K mod N and 3K mod N, you can easily compute its following elements.”

Show how this can be done. 9.12 Show how RSA can be represented by matrices M1, M2, and M3 of Problem 9.1. 9.13 Consider the following scheme: 1. Pick an odd number, E. 2. Pick two prime numbers, P and Q, where (P - 1)(Q - 1) -1 is evenly divisible by E. 3. Multiply P and Q to get N.

4. Calculate D =(P - 1)(Q - 1)(E - 1) + 1

E

Is this scheme equivalent to RSA? Show why or why not. 9.14 Consider the following scheme by which B encrypts a message for A. 1. A chooses two large primes P and Q that are also relatively prime to (P - 1)

and (Q - 1). 2. A publishes N = PQ as its public key.


3. A calculates P′and Q′ such that PP′ K 1 (mod Q - 1) and QQ′ K 1 (mod P - 1). 4. B encrypts message M as C = MN mod N. 5. A finds M by solving M K CP′

(mod Q) and M K CQ′(mod P).

a. Explain how this scheme works. b. How does it differ from RSA? c. Is there any particular advantage to RSA compared to this scheme? d. Show how this scheme can be represented by matrices M1, M2, and M3 of Problem 9.1. 9.15 “This is a very interesting case, Watson,” Holmes said. “The young man loves a girl,

and she loves him too. However, her father is a strange fellow who insists that his would-be son-in-law must design a simple and secure protocol for an appropriate public-key cryptosystem he could use in his company’s computer network. The young man came up with the following protocol for communication between two parties. For example, user A wishing to send message M to user B: (messages exchanged are in the format sender’s name, text, receiver’s name)”

1. A sends B the following block: (A, E(PUb, [M, A]), B). 2. B acknowledges receipt by sending to A the following block: (B, E(PUa, [M, B]), A).

“You can see that the protocol is really simple. But the girl’s father claims that the young man has not satisfied his call for a simple protocol, because the proposal con-tains a certain redundancy and can be further simplified to the following:”

1. A sends B the block: (A, E(PUb, M), B). 2. B acknowledges receipt by sending to A the block: (B, E(PUa, M), A).

“On the basis of that, the girl’s father refuses to allow his daughter to marry the young man, thus making them both unhappy. The young man was just here to ask me for help.”

“Hmm, I don’t see how you can help him.” Watson was visibly unhappy with the idea that the sympathetic young man has to lose his love.

“Well, I think I could help. You know, Watson, redundancy is sometimes good to ensure the security of protocol. Thus, the simplification the girl’s father has proposed could make the new protocol vulnerable to an attack the original protocol was able to resist,” mused Holmes. “Yes, it is so, Watson. Look, all an adversary needs is to be one of the users of the network and to be able to intercept messages exchanged between A and B. Being a user of the network, he has his own public encryption key and is able to send his own messages to A or to B and to receive theirs. With the help of the simplified protocol, he could then obtain message M user A has previously sent to B using the following procedure:”

Complete the description. 9.16 Use the fast exponentiation algorithm of Figure 9.8 to determine 5596 mod 1234.

Show the steps involved in the computation. 9.17 Here is another realization of the fast exponentiation algorithm. Demonstrate that it

is equivalent to the one in Figure 9.8. 1. f d 1; T d a; E d b 2. if odd(e) then f d f : T 3. E d [ E/2 ] 4. T d T : T 5. if E + 0 then goto 2 6. output f 9.18 The problem illustrates a simple application of the chosen ciphertext attack. Bob

intercepts a ciphertext C intended for Alice and encrypted with Alice’s public key e. Bob wants to obtain the original message M = Cd mod n. Bob chooses a random value r less than n and computes

Z = r e mod n X = ZC mod n t = r -1 mod n

appendix 9a / the Complexity of algorithmS 283

Next, Bob gets Alice to authenticate (sign) X with her private key (as in Figure 9.3), thereby decrypting X. Alice returns Y = Xd mod n. Show how Bob can use the infor-mation now available to him to determine M.

9.19 Show the OAEP decoding operation used for decryption that corresponds to the encoding operation of Figure 9.10.

9.20 Improve on algorithm P1 in Appendix 9A. a. Develop an algorithm that requires 2n multiplications and n + 1 additions. Hint:

xi+1 = xi * x. b. Develop an algorithm that requires only n + 1 multiplications and n + 1 additions.

Hint: P(x) = a0 + x * q(x), where q(x) is a polynomial of degree (n - 1).Note: The remaining problems concern the knapsack public-key algorithm described in Appendix J.

9.21 What items are in the knapsack in Figure F.1? 9.22 Perform encryption and decryption using the knapsack algorithm for the following: a. a′ = (1, 3, 5, 10); w = 7; m = 20; x = 1101 b. a′ = (1, 3, 5, 11, 23, 46, 136, 263); w = 203; m = 491; x = 11101000 c. a′ = (2, 3, 6, 12, 25); w = 46; m = 53; x = 11101 d. a′ = (15, 92, 108, 279, 563, 1172, 2243, 4468); w = 2393; m = 9291; x = 10110001

9.23 Why is it a requirement that m 7 an

1 = 1a′i?

aPPendix 9a the cOmPLexity Of aLgOrithms

The central issue in assessing the resistance of an encryption algorithm to crypt-analysis is the amount of time that a given type of attack will take. Typically, one cannot be sure that one has found the most efficient attack algorithm. The most that one can say is that, for a particular algorithm, the level of effort for an attack is of a particular order of magnitude. One can then compare that order of magnitude to the speed of current or predicted processors to determine the level of security of a particular algorithm.

A common measure of the efficiency of an algorithm is its time complexity. We define the time complexity of an algorithm to be f(n) if, for all n and all inputs of length n, the execution of the algorithm takes at most f(n) steps. Thus, for a given size of input and a given processor speed, the time complexity is an upper bound on the execution time.

There are several ambiguities here. First, the definition of a step is not precise. A step could be a single operation of a Turing machine, a single processor machine in-struction, a single high-level language machine instruction, and so on. However, these various definitions of step should all be related by simple multiplicative constants. For very large values of n, these constants are not important. What is important is how fast the relative execution time is growing. For example, if we are concerned about whether to use 50-digit (n = 1050) or 100-digit (n = 10100) keys for RSA, it is not nec-essary (or really possible) to know exactly how long it would take to break each size of key. Rather, we are interested in ballpark figures for level of effort and in knowing how much extra relative effort is required for the larger key size.

A second issue is that, generally speaking, we cannot pin down an exact for-mula for f(n). We can only approximate it. But again, we are primarily interested in the rate of change of f(n) as n becomes very large.


There is a standard mathematical notation, known as the “big-O” notation, for characterizing the time complexity of algorithms that is useful in this context. The definition is as follows: f(n) = O(g(n)) if and only if there exist two numbers a and M such that

0 f(n) 0 … a * 0 g(n) 0 , n Ú M (9.3)

An example helps clarify the use of this notation. Suppose we wish to evaluate a general polynomial of the form

P(x) = anxn + an - 1xn - 1 + c + a1x + a0

The following simple algorithm is from [POHL81].

algorithm P1; n, i, j: integer; x, polyval: real; a, S: array [0..100] of real; begin read(x, n); for i := 0 upto n do begin S[i] := 1; read(a[i]); for j := 1 upto i do S[i] := x * S[i]; S[i] := a[i] * S[i] end; polyval := 0; for i := 0 upto n do polyval := polyval + S[i]; write (‘value at’, x, ’is’, polyval) end.

In this algorithm, each subexpression is evaluated separately. Each S[i] requires (i + 1) multiplications: i multiplications to compute S[i] and one to multi-ply by a[i]. Computing all n terms requires

an

i = 0(i + 1) =

(n + 2)(n + 1)

2

multiplications. There are also (n + 1) additions, which we can ignore relative to the much larger number of multiplications. Thus, the time complexity of this algorithm is f(n) = (n + 2)(n + 1)/2. We now show that f(n) = O(n2). From the def-inition of Equation (9.3), we want to show that for a = 1 and M = 4 the relationship holds for g(n) = n2. We do this by induction on n. The relationship holds for n = 4 because (4 + 2) (4 + 1)/2 = 15 6 42 = 16. Now assume that it holds for all values of n up to k [i.e., (k + 2)(k + 1)/2 6 k2]. Then, with n = k + 1,

(n + 2)(n + 1)

2 =

(k + 3)(k + 2)

2

=(k + 2)(k + 1)

2 + k + 2

… k2 + k + 2

… k2 + 2k + 1 = (k + 1)2 = n2

Therefore, the result is true for n = k + 1.

appendix 9a / the Complexity of algorithmS 285

In general, the big-O notation makes use of the term that grows the fastest. For example,

1. O[ax7 + 3x3 + sin(x)] = O(ax7) = O(x7)

2. O(en + an10) = O(en)

3. O(n! + n50) = O(n!)

There is much more to the big-O notation, with fascinating ramifications. For the interested reader, two of the best accounts are in [GRAH94] and [KNUT97].

An algorithm with an input of size n is said to be

• Linear: If the running time is O(n)

• Polynomial: If the running time is O(nt) for some constant t

• Exponential: If the running time is O(th(n)) for some constant t and polynomial h(n)

Generally, a problem that can be solved in polynomial time is considered fea-sible, whereas anything worse than polynomial time, especially exponential time, is considered infeasible. But you must be careful with these terms. First, if the size of the input is small enough, even very complex algorithms become feasible. Suppose, for example, that you have a system that can execute 1012 operations per unit time. Table 9.6 shows the size of input that can be handled in one time unit for algorithms of various complexities. For algorithms of exponential or factorial time, only very small inputs can be accommodated.

The second thing to be careful about is the way in which the input is character-ized. For example, the complexity of cryptanalysis of an encryption algorithm can be characterized equally well in terms of the number of possible keys or the length of the key. For the Advanced Encryption Standard (AES), for example, the number of possible keys is 2128, and the length of the key is 128 bits. If we consider a single encryption to be a “step” and the number of possible keys to be N = 2n, then the time complexity of the algorithm is linear in terms of the number of keys [O(N)] but exponential in terms of the length of the key [O(2n)].

Table 9.6 Level of Effort for Various Levels of Complexity

Complexity size Operations

log2n 21012= 103 * 1011 1012

N 1012 1012

n2 106 1012

n6 102 1012

2n 39 1012

n! 15 1012

10.1 Diffie-Hellman Key Exchange

The AlgorithmKey Exchange ProtocolsMan-in-the-Middle Attack

10.2 Elgamal Cryptographic System

10.3 Elliptic Curve Arithmetic

Abelian GroupsElliptic Curves over Real NumbersElliptic Curves over ZpElliptic Curves over GF(2m)

10.4 Elliptic Curve Cryptography

Analog of Diffie-Hellman Key ExchangeElliptic Curve Encryption/DecryptionSecurity of Elliptic Curve Cryptography

10.5 Pseudorandom Number Generation Based on an Asymmetric Cipher

PRNG Based on RSAPRNG Based on Elliptic Curve Cryptography



Chapter

Other Public-Key cryPtOsystems

286286

10.1 / Diffie-Hellman Key excHange 287

Amongst the tribes of Central Australia every man, woman, and child has a secret or sacred name which is bestowed by the older men upon him or her soon after birth, and which is known to none but the fully initiated members of the group. This secret name is never mentioned except upon the most solemn occasions; to utter it in the hearing of men of another group would be a most serious breach of tribal custom. When mentioned at all, the name is spoken only in a whisper, and not until the most elaborate precautions have been taken that it shall be heard by no one but members of the group. The native thinks that a stranger knowing his secret name would have special power to work him ill by means of magic.


Learning Objectives


u Define Diffie-Hellman key exchange.

u Understand the man-in-the-middle attack.

u Present an overview of the Elgamal cryptographic system.

u Understand elliptic curve arithmetic.

u Present an overview of elliptic curve cryptography.

u Present two techniques for generating pseudorandom numbers using an asymmetric cipher.

This chapter begins with a description of one of the earliest and simplest PKCS: Diffie-Hellman key exchange. The chapter then looks at another important scheme, the Elgamal PKCS. Next, we look at the increasingly important PKCS known as elliptic curve cryptography. Finally, the use of public-key algorithms for pseudorandom num-ber generation is examined.

10.1 Diffie-HeLLman Key excHange

The first published public-key algorithm appeared in the seminal paper by Diffie and Hellman that defined public-key cryptography [DIFF76b] and is generally referred to as Diffie-Hellman key exchange.1 A number of commercial products employ this key exchange technique.

The purpose of the algorithm is to enable two users to securely exchange a key that can then be used for subsequent symmetric encryption of messages. The algorithm itself is limited to the exchange of secret values.

1Williamson of Britain’s CESG published the identical scheme a few months earlier in a classified docu-ment [WILL76] and claims to have discovered it several years prior to that; see [ELLI99] for a discussion.

288 cHapter 10 / OtHer public-Key cryptOsystems

The Diffie-Hellman algorithm depends for its effectiveness on the difficulty of computing discrete logarithms. Briefly, we can define the discrete logarithm in the following way. Recall from Chapter 8 that a primitive root of a prime number p is one whose powers modulo p generate all the integers from 1 to p - 1. That is, if a is a primitive root of the prime number p, then the numbers

a mod p, a2 mod p, c , ap - 1 mod p

are distinct and consist of the integers from 1 through p - 1 in some permutation.For any integer b and a primitive root a of prime number p, we can find a

unique exponent i such that

b K ai (mod p) where 0 … i … (p - 1)

The exponent i is referred to as the discrete logarithm of b for the base a, mod p. We express this value as dloga,p(b). See Chapter 8 for an extended discussion of discrete logarithms.

The Algorithm

Figure 10.1 summarizes the Diffie-Hellman key exchange algorithm. For this scheme, there are two publicly known numbers: a prime number q and an integer a that is a primitive root of q. Suppose the users A and B wish to create a shared key.

Alice Bob

Alice and Bob share aprime number q and aninteger α , such that α < q andα is a primitive root of q

Alice generates a privatekey XA such that XA < q

Alice calculates a publickey YA = αXA mod q

Alice receives Bob’spublic key YB in plaintext

Alice calculates sharedsecret key K = (YB)XA mod q

Bob calculates sharedsecret key K = (YA)XB mod q

Bob receives Alice’spublic key YA in plaintext

Bob calculates a publickey YB = αXB mod q

Bob generates a privatekey XB such that XB < q

Alice and Bob share aprime number q and aninteger α , such that α < q andα is a primitive root of q

YA YB

Figure 10.1 The Diffie-Hellman Key Exchange


User A selects a random integer XA 6 q and computes YA = aXA mod q. Similarly, user B independently selects a random integer XB 6 q and computes YB = aXB mod q. Each side keeps the X value private and makes the Y value avail-able publicly to the other side. Thus, XA is A’s private key and YA is A’s correspond-ing public key, and similarly for B. User A computes the key as K = (YB)XA mod q and user B computes the key as K = (YA)XB mod q. These two calculations produce identical results:

K = (YB)XA mod q

= (aXB mod q)XA mod q

= (aXB)XA mod q by the rules of modular arithmetic

= aXBXA mod q

= (aXA)XB mod q

= (aXA mod q)XB mod q

= (YA)XB mod q

The result is that the two sides have exchanged a secret value. Typically, this secret value is used as shared symmetric secret key. Now consider an adversary who can observe the key exchange and wishes to determine the secret key K. Because XA and XB are private, an adversary only has the following ingredients to work with: q, a, YA, and YB. Thus, the adversary is forced to take a discrete logarithm to determine the key. For example, to determine the private key of user B, an adversary must compute

XB = dloga,q(YB)

The adversary can then calculate the key K in the same manner as user B calculates it. That is, the adversary can calculate K as

K = (YA)XB mod q

The security of the Diffie-Hellman key exchange lies in the fact that, while it is relatively easy to calculate exponentials modulo a prime, it is very difficult to calculate discrete logarithms. For large primes, the latter task is considered infeasible.

Here is an example. Key exchange is based on the use of the prime number q = 353 and a primitive root of 353, in this case a = 3. A and B select private keys XA = 97 and XB = 233, respectively. Each computes its public key:

A computes YA = 397 mod 353 = 40.

B computes YB = 3233 mod 353 = 248.

After they exchange public keys, each can compute the common secret key:

A computes K = (YB)XA mod 353 = 24897 mod 353 = 160.

B computes K = (YA)XB mod 353 = 40233 mod 353 = 160.

We assume an attacker would have available the following information:

q = 353; a = 3; YA = 40; YB = 248


In this simple example, it would be possible by brute force to determine the secret key 160. In particular, an attacker E can determine the common key by discover-ing a solution to the equation 3a mod 353 = 40 or the equation 3b mod 353 = 248. The brute-force approach is to calculate powers of 3 modulo 353, stopping when the result equals either 40 or 248. The desired answer is reached with the exponent value of 97, which provides 397 mod 353 = 40.

With larger numbers, the problem becomes impractical.

Key Exchange Protocols

Figure 10.1 shows a simple protocol that makes use of the Diffie-Hellman calcula-tion. Suppose that user A wishes to set up a connection with user B and use a secret key to encrypt messages on that connection. User A can generate a one-time pri-vate key XA, calculate YA, and send that to user B. User B responds by generating a private value XB, calculating YB, and sending YB to user A. Both users can now calculate the key. The necessary public values q and a would need to be known ahead of time. Alternatively, user A could pick values for q and a and include those in the first message.

As an example of another use of the Diffie-Hellman algorithm, suppose that a group of users (e.g., all users on a LAN) each generate a long-lasting private value Xi (for user i) and calculate a public value Yi. These public values, together with global public values for q and a, are stored in some central directory. At any time, user j can access user i’s public value, calculate a secret key, and use that to send an encrypted message to user A. If the central directory is trusted, then this form of communication provides both confidentiality and a degree of authentication. Because only i and j can determine the key, no other user can read the message (confidentiality). Recipient i knows that only user j could have created a message using this key (authentication). However, the technique does not protect against replay attacks.

Man-in-the-Middle Attack

The protocol depicted in Figure 10.1 is insecure against a man-in-the-middle attack. Suppose Alice and Bob wish to exchange keys, and Darth is the adversary. The attack proceeds as follows (Figure 10.2).

1. Darth prepares for the attack by generating two random private keys XD1 and XD2 and then computing the corresponding public keys YD1 and YD2.

2. Alice transmits YA to Bob.

3. Darth intercepts YA and transmits YD1 to Bob. Darth also calculates K2 = (YA)XD2 mod q.

4. Bob receives YD1 and calculates K1 = (YD1)XB mod q.

5. Bob transmits YB to Alice.

6. Darth intercepts YB and transmits YD2 to Alice. Darth calculates K1 = (YB)XD1 mod q.

7. Alice receives YD2 and calculates K2 = (YD2)XA mod q.


At this point, Bob and Alice think that they share a secret key, but instead Bob and Darth share secret key K1 and Alice and Darth share secret key K2. All future communication between Bob and Alice is compromised in the following way.

1. Alice sends an encrypted message M: E(K2, M).

2. Darth intercepts the encrypted message and decrypts it to recover M.

3. Darth sends Bob E(K1, M) or E(K1, M′), where M′ is any message. In the first case, Darth simply wants to eavesdrop on the communication without altering it. In the second case, Darth wants to modify the message going to Bob.

The key exchange protocol is vulnerable to such an attack because it does not authenticate the participants. This vulnerability can be overcome with the use of digital signatures and public-key certificates; these topics are explored in Chapters 13 and 14.

Alice Darth Bob

Private key XAPublic keyYA = αXA mod q

Private key XBPublic keyYB = αXB mod q

Private keys XD1, XD2Public keysYD1 = αXD1 mod qYD2 = αXD2 mod q

YA

Secret keyK2 = (YA)XD2 mod q

Secret keyK1 = (YB)XD1 mod q

Secret keyK1 = (YD1)XB mod q

Secret keyK2 = (YD2)XA mod q

Alice and Darthshare K2

Bob and Darthshare K1

YD2 YD1

YB

Figure 10.2 Man-in-the-Middle Attack


10.2 eLgamaL cryptOgrapHic system

In 1984, T. Elgamal announced a public-key scheme based on discrete logarithms, closely related to the Diffie-Hellman technique [ELGA84, ELGA85]. The Elgamal2 cryptosystem is used in some form in a number of standards including the digital signature standard (DSS), which is covered in Chapter 13, and the S/MIME e-mail standard (Chapter 19).

As with Diffie-Hellman, the global elements of Elgamal are a prime number q and a, which is a primitive root of q. User A generates a private/public key pair as follows:

1. Generate a random integer XA, such that 1 6 XA 6 q - 1.

2. Compute Y A = aXA mod q.

3. A’s private key is XA and A’s public key is {q, a, YA}.

Any user B that has access to A’s public key can encrypt a message as follows:

1. Represent the message as an integer M in the range 0 … M … q - 1. Longer messages are sent as a sequence of blocks, with each block being an integer less than q.

2. Choose a random integer k such that 1 … k … q - 1.

3. Compute a one-time key K = (YA)k mod q.

4. Encrypt M as the pair of integers (C1, C2) where

C1 = ak mod q; C2 = KM mod q

User A recovers the plaintext as follows:

1. Recover the key by computing K = (C1)XA mod q.

2. Compute M = (C2K-1) mod q.

These steps are summarized in Figure 10.3. It corresponds to Figure 9.1a: Alice generates a public/private key pair; Bob encrypts using Alice’s public key; and Alice decrypts using her private key.

Let us demonstrate why the Elgamal scheme works. First, we show how K is recovered by the decryption process:

K = (YA)k mod q K is defined during the encryption process

K = (aXA mod q)k mod q substitute using YA = aXA mod q

K = akXA mod q by the rules of modular arithmetic

K = (C1)XA mod q substitute using C1 = ak mod q

Next, using K, we recover the plaintext as

C2 = KM mod q

(C2K-1) mod q = KMK-1 mod q = M mod q = M

2For no apparent reason, most of the literature uses the term ElGamal, although Mr. Elgamal’s last name does not have a capital letter G.

10.2 / elgamal cryptOgrapHic system 293

We can restate the Elgamal process as follows, using Figure 10.3.

1. Bob generates a random integer k.

2. Bob generates a one-time key K using Alice’s public-key components YA, q, and k.

3. Bob encrypts k using the public-key component a, yielding C1. C1 provides sufficient information for Alice to recover K.

4. Bob encrypts the plaintext message M using K.

5. Alice recovers K from C1 using her private key.

6. Alice uses K-1 to recover the plaintext message from C2.

Global Public Elements

q prime number

a a 6 q and a a primitive root of q

Key Generation by Alice

Select private XA XA 6 q - 1

Calculate YA YA = aXA mod q

Public key {q, a, YA}

Private key XA

Encryption by Bob with Alice’s Public Key

Plaintext: M 6 q

Select random integer k k 6 q

Calculate K K = (YA)k mod q

Calculate C1 C1 = ak mod q

Calculate C2 C2 = KM mod q

Ciphertext: (C1, C2)

Decryption by Alice with Alice’s Private Key

Ciphertext: (C1, C2)

Calculate K K = (C1)XA mod q

Plaintext: M = (C2K-1) mod q

Figure 10.3 The Elgamal Cryptosystem


Thus, K functions as a one-time key, used to encrypt and decrypt the message.

For example, let us start with the prime field GF(19); that is, q = 19. It has primitive roots {2, 3, 10, 13, 14, 15}, as shown in Table 8.3. We choose a = 10.

Alice generates a key pair as follows:

1. Alice chooses XA = 5.

2. Then YA = aXA mod q = a5 mod 19 = 3 (see Table 8.3).

3. Alice’s private key is 5 and Alice’s public key is {q, a, YA} = {19, 10, 3}.

Suppose Bob wants to send the message with the value M = 17. Then:

1. Bob chooses k = 6.

2. Then K = (YA)k mod q = 36 mod 19 = 729 mod 19 = 7.

3. So

C1 = ak mod q = a6 mod 19 = 11

C2 = KM mod q = 7 * 17 mod 19 = 119 mod 19 = 5

4. Bob sends the ciphertext (11, 5).

For decryption:

1. Alice calculates K = (C1)XA mod q = 115 mod 19 = 161051 mod 19 = 7.

2. Then K -1 in GF(19) is 7-1 mod 19 = 11.

3. Finally, M = (C2K- 1) mod q = 5 * 11 mod 19 = 55 mod 19 = 17.

If a message must be broken up into blocks and sent as a sequence of encrypted blocks, a unique value of k should be used for each block. If k is used for more than one block, knowledge of one block M1 of the message enables the user to compute other blocks as follows. Let

C1,1 = ak mod q; C2,1 = KM1 mod q

C1,2 = ak mod q; C2,2 = KM2 mod q

Then,C2,1

C2,2=

KM1 mod qKM2 mod q

=M1 mod qM2 mod q

If M1 is known, then M2 is easily computed as

M2 = (C2,1)-1 C2,2 M1 mod q

The security of Elgamal is based on the difficulty of computing discrete logarithms. To recover A’s private key, an adversary would have to compute XA = dloga,q(YA). Alternatively, to recover the one-time key K, an adversary would have to determine the random number k, and this would require computing the discrete logarithm k = dloga,q(C1). [STIN06] points out that these calculations are regarded as infeasible if p is at least 300 decimal digits and q - 1 has at least one “large” prime factor.

10.3 / elliptic curve aritHmetic 295

10.3 eLLiptic curve aritHmetic

Most of the products and standards that use public-key cryptography for encryption and digital signatures use RSA. As we have seen, the key length for secure RSA use has increased over recent years, and this has put a heavier processing load on applications using RSA. This burden has ramifications, especially for electronic com-merce sites that conduct large numbers of secure transactions. A competing system challenges RSA: elliptic curve cryptography (ECC). ECC is showing up in standard-ization efforts, including the IEEE P1363 Standard for Public-Key Cryptography.

The principal attraction of ECC, compared to RSA, is that it appears to offer equal security for a far smaller key size, thereby reducing processing overhead. On the other hand, although the theory of ECC has been around for some time, it is only recently that products have begun to appear and that there has been sustained cryptanalytic interest in probing for weaknesses. Accordingly, the confidence level in ECC is not yet as high as that in RSA.

ECC is fundamentally more difficult to explain than either RSA or Diffie-Hellman, and a full mathematical description is beyond the scope of this book. This section and the next give some background on elliptic curves and ECC. We begin with a brief review of the concept of abelian group. Next, we examine the concept of elliptic curves defined over the real numbers. This is followed by a look at elliptic curves defined over finite fields. Finally, we are able to examine elliptic curve ciphers.

The reader may wish to review the material on finite fields in Chapter 4 before proceeding.

Abelian Groups

Recall from Chapter 4 that an abelian group G, sometimes denoted by {G, # }, is a set of elements with a binary operation, denoted by # , that associates to each ordered pair (a, b) of elements in G an element (a # b) in G, such that the following axioms are obeyed:3

(A1) Closure: If a and b belong to G, then a # b is also in G.

(A2) Associative: a # (b # c) = (a # b) # c for all a, b, c in G.

(A3) Identity element: There is an element e in G such that a # e = e # a = a for all a in G.

(A4) Inverse element: For each a in G there is an element a′ in G such that a # a′ = a′ # a = e.

(A5) Commutative: a # b = b # a for all a, b in G.

A number of public-key ciphers are based on the use of an abelian group. For example, Diffie-Hellman key exchange involves multiplying pairs of nonzero integers modulo a prime number q. Keys are generated by exponentiation over

3The operator # is generic and can refer to addition, multiplication, or some other mathematical operation.


the group, with exponentiation defined as repeated multiplication. For example, ak mod q = (a * a * c * a) mod q. To attack Diffie-Hellman, the attacker must

k timesdetermine k given a and ak; this is the discrete logarithm problem.

For elliptic curve cryptography, an operation over elliptic curves, called addi-tion, is used. Multiplication is defined by repeated addition. For example,

a * k = (a + a + c + a)

k times

where the addition is performed over an elliptic curve. Cryptanalysis involves deter-mining k given a and (a * k).

An elliptic curve is defined by an equation in two variables with coefficients. For cryptography, the variables and coefficients are restricted to elements in a finite field, which results in the definition of a finite abelian group. Before looking at this, we first look at elliptic curves in which the variables and coefficients are real num-bers. This case is perhaps easier to visualize.

Elliptic Curves over Real Numbers

Elliptic curves are not ellipses. They are so named because they are described by cubic equations, similar to those used for calculating the circumference of an ellipse. In general, cubic equations for elliptic curves take the following form, known as a Weierstrass equation:

y2 + axy + by = x3 + cx2 + dx + e

where a, b, c, d, e are real numbers and x and y take on values in the real numbers.4 For our purpose, it is sufficient to limit ourselves to equations of the form

y2 = x3 + ax + b (10.1)

Such equations are said to be cubic, or of degree 3, because the highest exponent they contain is a 3. Also included in the definition of an elliptic curve is a single element denoted O and called the point at infinity or the zero point, which we discuss subsequently. To plot such a curve, we need to compute

y = 2x3 + ax + b

For given values of a and b, the plot consists of positive and negative values of y for each value of x. Thus, each curve is symmetric about y = 0. Figure 10.4 shows two examples of elliptic curves. As you can see, the formula sometimes produces weird-looking curves.

Now, consider the set of points E(a, b) consisting of all of the points (x, y) that satisfy Equation (10.1) together with the element O. Using a different value of the pair (a, b) results in a different set E(a, b). Using this terminology, the two curves in Figure 10.4 depict the sets E(-1, 0) and E(1, 1), respectively.

4Note that x and y are true variables, which take on values. This is in contrast to our discussion of polyno-mial rings and fields in Chapter 4, where x was treated as an indeterminate.

vv


Geometric Description of ADDition It can be shown that a group can be defined based on the set E(a, b) for specific values of a and b in Equation (10.1), provided the following condition is met:

4a3 + 27b2 ≠ 0 (10.2)

To define the group, we must define an operation, called addition and denoted by +, for the set E(a, b), where a and b satisfy Equation (10.2). In geometric terms, the rules for addition can be stated as follows: If three points on an elliptic curve lie on a straight line, their sum is O. From this definition, we can define the rules of addition over an elliptic curve.

�4

�2

0

2

4

�4

�2

0

2

4

543210�1�2

543210�1�2

(a) y2 � x3 � x

(b) y2 � x3 � x � 1

P

P

Q

Q

�(P � Q)

�(P � Q)

(P � Q)

(P � Q)

Figure 10.4 Example of Elliptic Curves


1. O serves as the additive identity. Thus O = -O; for any point P on the elliptic curve, P + O = P. In what follows, we assume P ≠ O and Q ≠ O.

2. The negative of a point P is the point with the same x coordinate but the nega-tive of the y coordinate; that is, if P = (x, y), then -P = (x, -y). Note that these two points can be joined by a vertical line. Note that P + (-P) = P - P = O.

3. To add two points P and Q with different x coordinates, draw a straight line between them and find the third point of intersection R. It is easily seen that there is a unique point R that is the point of intersection (unless the line is tangent to the curve at either P or Q, in which case we take R = P or R = Q, respectively). To form a group structure, we need to define addition on these three points: P + Q = -R. That is, we define P + Q to be the mirror image (with respect to the x axis) of the third point of intersection. Figure 10.4 illus-trates this construction.

4. The geometric interpretation of the preceding item also applies to two points, P and -P, with the same x coordinate. The points are joined by a vertical line, which can be viewed as also intersecting the curve at the infinity point. We therefore have P + (-P) = O, which is consistent with item (2).

5. To double a point Q, draw the tangent line and find the other point of inter-section S. Then Q + Q = 2Q = -S.

With the preceding list of rules, it can be shown that the set E(a, b) is an abe-lian group.

AlGebrAic Description of ADDition In this subsection, we present some results that enable calculation of additions over elliptic curves.5 For two distinct points, P = (xP , yP) and Q = (xQ, yQ), that are not negatives of each other, the slope of the line l that joins them is ∆ = (yQ - yP)>(xQ - xP). There is exactly one other point where l intersects the elliptic curve, and that is the negative of the sum of P and Q. After some algebraic manipulation, we can express the sum R = P + Q as

xR = ∆2 - xP - xQ (10.3)

yR = -yP + ∆(xP - xR)

We also need to be able to add a point to itself: P + P = 2P = R. When yP ≠ 0, the expressions are

xR = a3x2P + a

2yPb

2

-2xP

(10.4)

yR = a3xP2 + a

2yPb(xP - xR) - yP

Elliptic Curves over Zp

Elliptic curve cryptography makes use of elliptic curves in which the variables and coefficients are all restricted to elements of a finite field. Two families of elliptic curves are used in cryptographic applications: prime curves over Zp and binary

5For derivations of these results, see [KOBL94] or other mathematical treatments of elliptic curves.


curves over GF(2m). For a prime curve over Zp, we use a cubic equation in which the variables and coefficients all take on values in the set of integers from 0 through p - 1 and in which calculations are performed modulo p. For a binary curve de-fined over GF(2m), the variables and coefficients all take on values in GF(2m) and in calculations are performed over GF(2m). [FERN99] points out that prime curves are best for software applications, because the extended bit-fiddling operations needed by binary curves are not required; and that binary curves are best for hard-ware applications, where it takes remarkably few logic gates to create a powerful, fast cryptosystem. We examine these two families in this section and the next.

There is no obvious geometric interpretation of elliptic curve arithmetic over finite fields. The algebraic interpretation used for elliptic curve arithmetic over real numbers does readily carry over, and this is the approach we take.

For elliptic curves over Zp, as with real numbers, we limit ourselves to equa-tions of the form of Equation (10.1), but in this case with coefficients and variables limited to Zp:

y2 mod p = (x3 + ax + b) mod p (10.5)

For example, Equation (10.5) is satisfied for a = 1, b = 1, x = 9, y = 7, p = 23:

72 mod 23 = (93 + 9 + 1) mod 23

49 mod 23 = 739 mod 23

3 = 3

Now consider the set Ep(a, b) consisting of all pairs of integers (x, y) that sat-isfy Equation (10.5), together with a point at infinity O. The coefficients a and b and the variables x and y are all elements of Zp.

For example, let p = 23 and consider the elliptic curve y2 = x3 + x + 1. In this case, a = b = 1. Note that this equation is the same as that of Figure 10.4b. The figure shows a continuous curve with all of the real points that satisfy the equation. For the set E23(1, 1), we are only interested in the nonnegative integers in the quad-rant from (0, 0) through (p - 1, p - 1) that satisfy the equation mod p. Table 10.1 lists the points (other than O) that are part of E23(1, 1). Figure 10.5 plots the points of E23(1, 1); note that the points, with one exception, are symmetric about y = 11.5.

Table 10.1 Points (other than O) on the Elliptic Curve E23 (1,1)

(0, 1) (6, 4) (12, 19)

(0, 22) (6, 19) (13, 7)

(1, 7) (7, 11) (13, 16)

(1, 16) (7, 12) (17, 3)

(3, 10) (9, 7) (17, 20)

(3, 13) (9, 16) (18, 3)

(4, 0) (11, 3) (18, 20)

(5, 4) (11, 20) (19, 5)

(5, 19) (12, 4) (19, 18)


It can be shown that a finite abelian group can be defined based on the set Ep(a, b) provided that (x3 + ax + b) mod p has no repeated factors. This is equiva-lent to the condition

(4a3 + 27b2) mod p ≠ 0 mod p (10.6)

Note that Equation (10.6) has the same form as Equation (10.2).The rules for addition over Ep(a, b), correspond to the algebraic technique de-

scribed for elliptic curves defined over real numbers. For all points P, Q ∈ Ep(a, b):

1. P + O = P.

2. If P = (xP, yP), then P + (xP,-yP) = O. The point (xP,-yP) is the nega-tive of P, denoted as -P. For example, in E23(1, 1), for P = (13, 7), we have -P = (13, -7). But -7 mod 23 = 16. Therefore, -P = (13, 16), which is also in E23(1, 1).

3. If P = (xp, yp) and Q = (xQ, yQ) with P ≠ -Q, then R = P + Q = (xR, yR) is determined by the following rules:

xR = (l2 - xP - xQ) mod p

yR = (l(xP - xR) - yP) mod p

00

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

1 2 3 4 5 6 7 8 9 10 11x

y

12 13 14 15 16 17 18 19 20 21 22

Figure 10.5 The Elliptic Curve E23(1, 1)


where

l = e ayQ - yP

xQ - xPb mod p if P ≠ Q

a3xP

2 + a

2yP

b mod p if P = Q

4. Multiplication is defined as repeated addition; for example, 4P = P + P + P + P.

For example, let P = (3, 10) and Q = (9, 7) in E23(1, 1). Then

l = a7 - 109 - 3

bmod 23 = a -36

bmod 23 = a -12

bmod 23 = 11

xR = (112 - 3 - 9) mod 23 = 109 mod 23 = 17

yR = (11(3 - 17) - 10) mod 23 = -164 mod 23 = 20

So P + Q = (17, 20). To find 2P,

l = a3(32) + 1

2 * 10bmod 23 = a 5

20bmod 23 = a1

4bmod 23 = 6

The last step in the preceding equation involves taking the multiplicative in-verse of 4 in Z23. This can be done using the extended Euclidean algorithm defined in Section 4.4. To confirm, note that (6 * 4) mod 23 = 24 mod 23 = 1.

xR = (62 - 3 - 3) mod 23 = 30 mod 23 = 7

yR = (6(3 - 7) - 10) mod 23 = (-34) mod 23 = 12

and 2P = (7, 12).For determining the security of various elliptic curve ciphers, it is of some in-

terest to know the number of points in a finite abelian group defined over an elliptic curve. In the case of the finite group EP(a, b), the number of points N is bounded by

p + 1 - 22p … N … p + 1 + 22p

Note that the number of points in Ep(a, b) is approximately equal to the number of elements in Zp, namely p elements.

Elliptic Curves over GF(2m)

Recall from Chapter 4 that a finite field GF(2m) consists of 2m elements, together with addition and multiplication operations that can be defined over polynomials. For elliptic curves over GF(2m), we use a cubic equation in which the variables and coefficients all take on values in GF(2m) for some number m and in which calcula-tions are performed using the rules of arithmetic in GF(2m).

It turns out that the form of cubic equation appropriate for cryptographic ap-plications for elliptic curves is somewhat different for GF(2m) than for Zp. The form is

y2 + xy = x3 + ax2 + b (10.7)


where it is understood that the variables x and y and the coefficients a and b are ele-ments of GF(2m) and that calculations are performed in GF(2m).

Now consider the set E2m(a, b) consisting of all pairs of integers (x, y) that sat-isfy Equation (10.7), together with a point at infinity O.

For example, let us use the finite field GF(24) with the irreducible polynomial f(x) = x4 + x + 1. This yields a generator g that satisfies f(g) = 0 with a value of g4 = g + 1, or in binary, g = 0010. We can develop the powers of g as follows.

g0 = 0001 g4 = 0011 g8 = 0101 g12 = 1111

g1 = 0010 g5 = 0110 g9 = 1010 g13 = 1101

g2 = 0100 g6 = 1100 g10 = 0111 g14 = 1001

g3 = 1000 g7 = 1011 g11 = 1110 g15 = 0001

For example, g5 = (g4)(g) = (g + 1)(g) = g2 + g = 0110.Now consider the elliptic curve y2 + xy = x3 + g4x2 + 1. In this case, a = g4

and b = g0 = 1. One point that satisfies this equation is (g5, g3):

(g3)2 + (g5)(g3) = (g5)3 + (g4)(g5)2 + 1

g6 + g8 = g15 + g14 + 1

1100 + 0101 = 0001 + 1001 + 0001

1001 = 1001

Table 10.2 lists the points (other than O) that are part of E24(g4, 1). Figure 10.6 plots the points of E24(g4, 1).

It can be shown that a finite abelian group can be defined based on the set E2m(a, b), provided that b ≠ 0. The rules for addition can be stated as follows. For all points P, Q ∈ E2m(a, b):

1. P + O = P.

2. If P = (xP, yP), then P + (xP, xP + yP) = O. The point (xP, xP + yP) is the negative of P, which is denoted as -P.

3. If P = (xP, yP) and Q = (xQ, yQ) with P ≠ -Q and P ≠ Q, then R = P + Q = (xR, yR) is determined by the following rules:

xR = l2 + l + xP + xQ + a

yR = l(xP + xR) + xR + yP

Table 10.2 Points (other than O) on the Elliptic Curve E24(g4, 1)

(0, 1) (g5, g3) (g9, g13)

(1, g6) (g5, g11) (g10, g)

(1, g13) (g6, g8) (g10, g8)

(g3, g8) (g6, g14) (g12, 0)

(g3, g13) (g9, g10) (g12, g12)

10.4 / elliptic curve cryptOgrapHy 303

where

l =yQ + yP

xQ + xP

4. If P = (xP, yP) then R = 2P = (xR, yR) is determined by the following rules:

xR = l2 + l + a

yR = xP2 + (l + 1)xR

where

l = xP +yP

xP

10.4 eLLiptic curve cryptOgrapHy

The addition operation in ECC is the counterpart of modular multiplication in RSA, and multiple addition is the counterpart of modular exponentiation. To form a cryptographic system using elliptic curves, we need to find a “hard problem” cor-responding to factoring the product of two primes or taking the discrete logarithm.

Consider the equation Q = kP where Q, P ∈ EP(a, b) and k 6 p. It is rela-tively easy to calculate Q given k and P, but it is hard to determine k given Q and P. This is called the discrete logarithm problem for elliptic curves.

We give an example taken from the Certicom Web site (www.certicom .com). Consider the group E23(9,17). This is the group defined by the equation y2 mod 23 = (x3 + 9x + 17) mod 23. What is the discrete logarithm k of Q = (4, 5) to the base P = (16, 5)? The brute-force method is to compute multiples of P until

11

g

g2g3

g4

g5

g6

g7

g8

g9

g10

g11

g12

g13

g14

0

g g2 g3 g4 g5 g6 g7 g8 g9 g10 g11

x

y

g12 g13 g14 0

Figure 10.6 The Elliptic Curve E24 (g4, 1)

www.certicom.com

www.certicom.com


Q is found. Thus,

P = (16, 5); 2P = (20, 20); 3P = (14, 14); 4P = (19, 20); 5P = (13, 10);

6P = (7, 3); 7P = (8, 7); 8P = (12, 17); 9P = (4, 5)

Because 9P = (4, 5) = Q, the discrete logarithm Q = (4, 5) to the base P = (16, 5) is k = 9. In a real application, k would be so large as to make the brute-force approach infeasible.

In the remainder of this section, we show two approaches to ECC that give the flavor of this technique.

Analog of Diffie-Hellman Key Exchange

Key exchange using elliptic curves can be done in the following manner. First pick a large integer q, which is either a prime number p or an integer of the form 2m, and elliptic curve parameters a and b for Equation (10.5) or Equation (10.7). This defines the elliptic group of points Eq(a, b). Next, pick a base point G = (x1, y1) in Ep(a, b) whose order is a very large value n. The order n of a point G on an elliptic curve is the smallest positive integer n such that nG = 0 and G are parameters of the cryptosystem known to all participants.

A key exchange between users A and B can be accomplished as follows (Figure 10.7).

1. A selects an integer nA less than n. This is A’s private key. A then generates a public key PA = nA * G; the public key is a point in Eq(a, b).

2. B similarly selects a private key nB and computes a public key PB. 3. A generates the secret key k = nA * PB. B generates the secret key

k = nB * PA.

The two calculations in step 3 produce the same result because

nA * PB = nA * (nB * G) = nB * (nA * G) = nB * PA

To break this scheme, an attacker would need to be able to compute k given G and kG, which is assumed to be hard.

As an example,6 take p = 211; Ep(0,-4), which is equivalent to the curve y2 = x3 - 4; and G = (2, 2). One can calculate that 240G = O. A’s private key is nA = 121, so A’s public key is PA = 121(2, 2) = (115, 48). B’s private key is nB = 203, so B’s public key is 203(2, 3) = (130, 203). The shared secret key is 121(130, 203) = 203(115, 48) = (161, 69).

Note that the secret key is a pair of numbers. If this key is to be used as a ses-sion key for conventional encryption, then a single number must be generated. We could simply use the x coordinates or some simple function of the x coordinate.

Elliptic Curve Encryption/Decryption

Several approaches to encryption/decryption using elliptic curves have been ana-lyzed in the literature. In this subsection, we look at perhaps the simplest. The first task in this system is to encode the plaintext message m to be sent as an (x, y) point Pm.

6Provided by Ed Schaefer of Santa Clara University.

10.4 / elliptic curve cryptOgrapHy 305

It is the point Pm that will be encrypted as a ciphertext and subsequently decrypted. Note that we cannot simply encode the message as the x or y coordinate of a point, because not all such coordinates are in Eq(a, b); for example, see Table 10.1. Again, there are several approaches to this encoding, which we will not address here, but suffice it to say that there are relatively straightforward techniques that can be used.

As with the key exchange system, an encryption/decryption system requires a point G and an elliptic group Eq(a, b) as parameters. Each user A selects a private key nA and generates a public key PA = nA * G.

To encrypt and send a message Pm to B, A chooses a random positive integer k and produces the ciphertext Cm consisting of the pair of points:

Cm = {kG, Pm + kPB}

Note that A has used B’s public key PB. To decrypt the ciphertext, B multiplies the first point in the pair by B’s private key and subtracts the result from the second point:

Pm + kPB - nB(kG) = Pm + k(nBG) - nB(kG) = Pm

A has masked the message Pm by adding kPB to it. Nobody but A knows the value of k, so even though Pb is a public key, nobody can remove the mask kPB. However, A also includes a “clue,” which is enough to remove the mask if one

Global Public Elements

Eq(a, b) elliptic curve with parameters a, b, and q, where q is a prime or an integer of the form 2m

G point on elliptic curve whose order is large value n

User A Key Generation

Select private nA nA 6 n

Calculate public PA PA = nA * G

User B Key Generation

Select private nB nB 6 n

Calculate public PB PB = nB * G

Calculation of Secret Key by User A

K = nA * PB

Calculation of Secret Key by User B

K = nB * PA

Figure 10.7 ECC Diffie-Hellman Key Exchange


knows the private key nB. For an attacker to recover the message, the attacker would have to compute k given G and kG, which is assumed to be hard.

Let us consider a simple example. The global public elements are q = 257; Eq(a, b) = E257(0, -4), which is equivalent to the curve y2 = x3 - 4; and G = (2, 2). Bob’s private key is nB = 101, and his public key is PB = nBG = 101(2, 2) = (197, 167). Alice wishes to send a message to Bob that is encoded in the elliptic point Pm = (112, 26). Alice chooses random integer k = 41 and computes kG = 41(2, 2) = (136, 128), kPB = 41(197, 167) = (68, 84) and Pm + kPB = (112, 26) + (68, 84) = (246, 174). Alice sends the ciphertext Cm = (C1, C2) = {(136, 128), (246, 174)} to Bob. Bob receives the ciphertext and computes C2 - nBC1 = (246, 174) - 101(136, 128) = (246, 174) - (68, 84) = (112, 26).

Security of Elliptic Curve Cryptography

The security of ECC depends on how difficult it is to determine k given kP and P. This is referred to as the elliptic curve logarithm problem. The fastest known tech-nique for taking the elliptic curve logarithm is known as the Pollard rho method. Table 10.3, from NIST SP800-57 (Recommendation for Key Management—Part 1: General, July 2012), compares various algorithms by showing comparable key sizes in terms of computational effort for cryptanalysis. As can be seen, a considerably smaller key size can be used for ECC compared to RSA. Furthermore, for equal key lengths, the computational effort required for ECC and RSA is comparable [JURI97]. Thus, there is a computational advantage to using ECC with a shorter key length than a comparably secure RSA.

10.5 pseuDOranDOm number generatiOn baseD On an asymmetric cipHer

We noted in Chapter 7 that because a symmetric block cipher produces an appar-ently random output, it can serve as the basis of a pseudorandom number generator (PRNG). Similarly, an asymmetric encryption algorithm produces apparently random

Table 10.3 Comparable Key Sizes in Terms of Computational Effort for Cryptanalysis (NIST SP-800-57)

Symmetric Key Algorithms

Diffie-Hellman, Digital Signature Algorithm

RSA (size of n in bits)

ECC (modulus size in bits)

80L = 1024N = 160

1024 160–223

112L = 2048N = 224

2048 224–255

128L = 3072N = 256

3072 256–383

192L = 7680N = 384

7680 384–511

256L = 15,360N = 512

15,360 512+

Note: L = size of public key, N = size of private key

10.5 / pseuDOranDOm number generatiOn 307

output and can be used to build a PRNG. Because asymmetric algorithms are typically much slower than symmetric algorithms, asymmetric algorithms are not used to gener-ate open-ended PRNG bit streams. Rather, the asymmetric approach is useful for creat-ing a pseudorandom function (PRF) for generating a short pseudorandom bit sequence.

In this section, we examine two PRNG designs based on pseudorandom functions.

PRNG Based on RSA

For a sufficient key length, the RSA algorithm is considered secure and is a good candidate to form the basis of a PRNG. Such a PRNG, known as the Micali-Schnorr PRNG [MICA91], is recommended in the ANSI standard X9.82 (Random Number Generation) and in the ISO standard 18031 (Random Bit Generation).

The PRNG is illustrated in Figure 10.8. As can be seen, this PRNG has much the same structure as the output feedback (OFB) mode used as a PRNG (see Figure 7.4b and the portion of Figure 6.6a enclosed with a dashed box). In this case, the encryption algorithm is RSA rather than a symmetric block cipher. Also, a portion of the output is fed back to the next iteration of the encryption algorithm and the remainder of the output is used as pseudorandom bits. The motivation for this separation of the output into two distinct parts is so that the pseudorandom bits from one stage do not provide input to the next stage. This separation should contribute to forward unpredictability.

We can define the PRNG as follows.

Setup Select p, q primes; n = pq; f(n) = (p - 1)(q - 1). Select e such that gcd(e, f(n)) = 1. These are the standard RSA setup selections (see Figure 9.5). In addition, let N = [log2 n] + 1 (the bitlength of n). Select r, k such that r + k = N.

Seed Select a random seed x0 of bitlength r.

Generate Generate a pseudorandom sequence of length k * m using the loop for i from 1 to m do

yi = xi- 1e mod n

xi = r most significant bits of yi zi = k least significant bits of yi

Output The output sequence is z1 }z2 } c}zm.

Seed = x0

x1 = r mostsignificant bits

z1 = k leastsignificant bits

y1 = x0 mod ne

n, e, r, k n, e, r, k n, e, r, k





y2 = x1 mod ne y3 = x2 mod ne

Encrypt Encrypt Encrypt

Figure 10.8 Micali-Schnorr Pseudorandom Bit Generator


The parameters n, r, e, and k are selected to satisfy the following six requirements.

1. n = pq n is chosen as the product of two primes to have the cryptographic strength required of RSA.

2. 1 6 e 6 f(n); gcd (e, f (n)) = 1 Ensures that the mapping s S se mod n is 1 to 1.

3. re Ú 2N Ensures that the exponentiation requires a full modular reduction.

4. r Ú 2 strength Protects against a cryptographic attacks.

5. k, r are multiples of 8 An implementation convenience.

6. k Ú 8; r + k = N All bits are used.

The variable strength in requirement 4 is defined in NIST SP 800-90 as fol-lows: A number associated with the amount of work (that is, the number of opera-tions) required to break a cryptographic algorithm or system; a security strength is specified in bits and is a specific value from the set (112, 128, 192, 256) for this Recommendation. The amount of work needed is 2strength.

There is clearly a tradeoff between r and k. Because RSA is computationally intensive compared to a block cipher, we would like to generate as many pseudo-random bits per iteration as possible and therefore would like a large value of k. However, for cryptographic strength, we would like r to be as large as possible.

For example, if e = 3 and N = 1024, then we have the inequality 3r 7 1024, yielding a minimum required size for r of 683 bits. For r set to that size, k = 341 bits are generated for each exponentiation (each RSA encryption). In this case, each exponentiation requires only one modular squaring of a 683-bit number and one modular multiplication. That is, we need only calculate (xi * (xi

2 mod n)) mod n.

PRNG Based on Elliptic Curve Cryptography

In this subsection, we briefly summarize a technique developed by the U.S. National Security Agency (NSA) known as dual elliptic curve PRNG (DEC PRNG). This technique is recommended in NIST SP 800-90, the ANSI standard X9.82, and the ISO standard 18031. There has been some controversy regarding both the security and efficiency of this algorithm compared to other alternatives (e.g., see [SCHO06], [BROW07]).

[SCHO06] summarizes the algorithm as follows: Let P and Q be two known points on a given elliptic curve. The seed of the DEC PRNG is a random integer s0 ∈ {0, 1, c, ⧣E(GF(p)) - 1}, where ⧣ E(GF(p)) denotes the number of points on the curve. Let x denote a function that gives the x-coordinate of a point of the curve. Let lsbi(s) denote the i least significant bits of an integer s. The DEC PRNG transforms the seed into the pseudorandom sequence of length 240k, k 7 0, as follows.

10.7 / Key terms, review QuestiOns, anD prOblems 309

for i = 1 to k do

Set si d x(Si-1 P)

Set ri d lsb240 (x(si Q))

end for

Return r1,...,rk

Given the security concerns expressed for this PRNG, the only motivation for its use would be that it is used in a system that already implements ECC but does not implement any other symmetric, asymmetric, or hash cryptographic algorithm that could be used to build a PRNG.


A quite readable treatment of elliptic curve cryptography is [ROSI99]; the emphasis is on software implementation. Another readable, but rigorous, book is [HANK04]. There are also good but more concise descriptions in [KUMA98], [STIN06], and [KOBL94]. Two inter-esting survey treatments are [FERN99] and [JURI97].

FERN99 Fernandes, A. “Elliptic Curve Cryptography.” Dr. Dobb’s Journal, December 1999.

HANK04 Hankerson, D.; Menezes, A.; and Vanstone, S. Guide to Elliptic Curve Cryptography. New York: Springer, 2004.

JURI97 Jurisic, A., and Menezes, A. “Elliptic Curves and Cryptography.” Dr. Dobb’s Journal, April 1997.

KOBL94 Koblitz, N. A Course in Number Theory and Cryptography. New York: Springer-Verlag, 1994.

KUMA98 Kumanduri, R., and Romero, C. Number Theory with Computer Applications. Upper Saddle River, NJ: Prentice Hall, 1998.

ROSI99 Rosing, M. Implementing Elliptic Curve Cryptography. Greeenwich, CT: Manning Publications, 1999.

STIN06 Stinson, D. Cryptography: Theory and Practice. Boca Raton, FL: CRC Press, 2006.


Key Terms

abelian groupbinary curvecubic equationDiffie-Hellman key exchangediscrete logarithm

elliptic curveelliptic curve arithmeticelliptic curve cryptographyfinite fieldman-in-the-middle attack

Micali-Schnorrprime curveprimitive rootzero point


Review Questions 10.1 Briefly explain Diffie-Hellman key exchange. 10.2 What is an elliptic curve? 10.3 What is the zero point of an elliptic curve? 10.4 What is the sum of three points on an elliptic curve that lie on a straight line?

Problems 10.1 Users A and B use the Diffie-Hellman key exchange technique with a common prime

q = 71 and a primitive root a = 7. a. If user A has private key XA = 5, what is A’s public key YA? b. If user B has private key XB = 12, what is B’s public key YB? c. What is the shared secret key? 10.2 Consider a Diffie-Hellman scheme with a common prime q = 11 and a primitive root

a = 2. a. Show that 2 is a primitive root of 11. b. If user A has public key YA = 9, what is A’s private key XA? c. If user B has public key YB = 3, what is the secret key K shared with A? 10.3 In the Diffie-Hellman protocol, each participant selects a secret number x and sends

the other participant ax mod q for some public number a. What would happen if the participants sent each other xa for some public number a instead? Give at least one method Alice and Bob could use to agree on a key. Can Eve break your system with-out finding the secret numbers? Can Eve find the secret numbers?

10.4 This problem illustrates the point that the Diffie-Hellman protocol is not secure with-out the step where you take the modulus; i.e. the “Indiscrete Log Problem” is not a hard problem! You are Eve and have captured Alice and Bob and imprisoned them. You overhear the following dialog.

Bob: Oh, let’s not bother with the prime in the Diffie-Hellman protocol, it will make things easier.

Alice: Okay, but we still need a base a to raise things to. How about a = 3?

Bob: All right, then my result is 27.

Alice: And mine is 243.

What is Bob’s private key XB and Alice’s private key XA? What is their secret com-bined key? (Don’t forget to show your work.)

10.5 Section 10.1 describes a man-in-the-middle attack on the Diffie-Hellman key ex-change protocol in which the adversary generates two public–private key pairs for the attack. Could the same attack be accomplished with one pair? Explain.

10.6 Consider an Elgamal scheme with a common prime q = 71 and a primitive root a = 7.

a. If B has public key YB = 3 and A choose the random integer k = 2, what is the ciphertext of M = 30?

b. If A now chooses a different value of k so that the encoding of M = 30 is C = (59, C2), what is the integer C2?

10.7 Rule (5) for doing arithmetic in elliptic curves over real numbers states that to double a point Q2, draw the tangent line and find the other point of intersection S. Then Q + Q = 2Q = -S. If the tangent line is not vertical, there will be exactly one point of intersection. However, suppose the tangent line is vertical? In that case, what is the value 2Q? What is the value 3Q?

10.7 / Key terms, review QuestiOns, anD prOblems 311

10.8 Demonstrate that the two elliptic curves of Figure 10.4 each satisfy the conditions for a group over the real numbers.

10.9 Is (4, 7) a point on the elliptic curve y2 = x3 - 5x + 5 over real numbers? 10.10 On the elliptic curve over the real numbers y2 = x3 - 36x, let P = (-3.5, 9.5) and

Q = (-2.5, 8.5). Find P + Q and 2P. 10.11 Does the elliptic curve equation y2 = x3 + 10x + 5 define a group over Z17? 10.12 Consider the elliptic curve E11(1, 6); that is, the curve is defined by y2 = x3 + x + 6

with a modulus of p = 11. Determine all of the points in E11(1, 6). Hint: Start by cal-culating the right-hand side of the equation for all values of x.

10.13 What are the negatives of the following elliptic curve points over Z17? P = (5, 8); Q = (3, 0); R = (0, 6).

10.14 For E11(1, 6), consider the point G = (2, 7). Compute the multiples of G from 2G through 13G.

10.15 This problem performs elliptic curve encryption/decryption using the scheme out-lined in Section 10.4. The cryptosystem parameters are E11(1, 6) and G = (2, 7). B’s private key is nB = 7.

a. Find B’s public key PB. b. A wishes to encrypt the message Pm = (10, 9) and chooses the random value

k = 3. Determine the ciphertext Cm. c. Show the calculation by which B recovers Pm from Cm. 10.16 The following is a first attempt at an elliptic curve signature scheme. We have a global

elliptic curve, prime p, and “generator” G. Alice picks a private signing key XA and forms the public verifying key YA = XAG. To sign a message M:

• Alice picks a value k. • Alice sends Bob M, k and the signature S = M - kXAG. • Bob verifies that M = S + kYA. a. Show that this scheme works. That is, show that the verification process produces

an equality if the signature is valid. b. Show that the scheme is unacceptable by describing a simple technique for forging

a user’s signature on an arbitrary message. 10.17 Here is an improved version of the scheme given in the previous problem. As before,

we have a global elliptic curve, prime p, and “generator” G. Alice picks a private signing key XA and forms the public verifying key YA = XAG. To sign a message M:

• Bob picks a value k. • Bob sends Alice C1 = kG. • Alice sends Bob M and the signature S = M - XAC1. • Bob verifies that M = S + kYA. a. Show that this scheme works. That is, show that the verification process produces

an equality if the signature is valid. b. Show that forging a message in this scheme is as hard as breaking (Elgamal)

elliptic curve cryptography. (Or find an easier way to forge a message?) c. This scheme has an extra “pass” compared to other cryptosystems and signature

schemes we have looked at. What are some drawbacks to this?


Part 3: CryPtograPhiC Data integrity algorithms

Chapter

CryptographiC hash FunCtions

11.1 Applications of Cryptographic Hash Functions

Message AuthenticationDigital SignaturesOther Applications

11.2 Two Simple Hash Functions

11.3 Requirements and Security

Security Requirements for Cryptographic Hash FunctionsBrute-Force AttacksCryptanalysis

11.4 Hash Functions Based on Cipher Block Chaining

11.5 Secure Hash Algorithm (SHA)

SHA-512 LogicSHA-512 Round FunctionExample

11.6 SHA-3

The Sponge ConstructionThe SHA-3 Iteration Function f



313

314 Chapter 11 / CryptographiC hash FunCtions

“The fish that you have tattooed immediately above your right wrist could only have been done in China. I have made a small study of tattoo marks and have even contributed to the literature on the subject.”

—The Red-Headed League, Sir Arthur Conan Doyle

The Douglas Squirrel has a distinctive eating habit. It usually eats pine cones from the bottom end up. Partially eaten cones can indicate the presence of these squirrels if they have been attacked from the bottom first. If, instead, the cone has been eaten from the top end down, it is more likely to have been a crossbill finch that has been doing the dining.

—Squirrels: A Wildlife Handbook, Kim Long

A hash function H accepts a variable-length block of data M as input and produces a fixed-size hash value h = H(M). A “good” hash function has the property that the results of applying the function to a large set of inputs will produce outputs that are evenly distributed and apparently random. In general terms, the principal object of a hash function is data integrity. A change to any bit or bits in M results, with high prob-ability, in a change to the hash code.

The kind of hash function needed for security applications is referred to as a cryptographic hash function. A cryptographic hash function is an algorithm for which it is computationally infeasible (because no attack is significantly more efficient than brute force) to find either (a) a data object that maps to a pre-specified hash result (the one-way property) or (b) two data objects that map to the same hash result (the collision-free property). Because of these characteristics, hash functions are often used to determine whether or not data has changed.

Learning Objectives


u Summarize the applications of cryptographic hash functions.

u Explain why a hash function used for message authentication needs to be secured.

u Understand the differences among preimage resistant, second preimage resistant, and collision resistant properties.

u Present an overview of the basic structure of cryptographic hash functions.

u Describe how cipher block chaining can be used to construct a hash function.

u Understand the operation of SHA-512.

u Understand the birthday paradox and present an overview of the birthday attack.

11.1 / appliCations oF CryptographiC hash FunCtions 315

Figure 11.1 depicts the general operation of a cryptographic hash func-tion. Typically, the input is padded out to an integer multiple of some fixed length (e.g., 1024 bits), and the padding includes the value of the length of the original mes-sage in bits. The length field is a security measure to increase the difficulty for an attacker to produce an alternative message with the same hash value.

This chapter begins with a discussion of the wide variety of applications for cryp-tographic hash functions. Next, we look at the security requirements for such functions. Then we look at the use of cipher block chaining to implement a cryptographic hash function. The remainder of the chapter is devoted to the most important and widely used family of cryptographic hash functions, the Secure Hash Algorithm (SHA) family.

Appendix describes MD5, a well-known cryptographic hash function with similarities to SHA-1.

11.1 appLicatiOns Of cryptOgraphic hash functiOns

Perhaps the most versatile cryptographic algorithm is the cryptographic hash func-tion. It is used in a wide variety of security applications and Internet protocols. To better understand some of the requirements and security implications for crypto-graphic hash functions, it is useful to look at the range of applications in which it is employed.

Message Authentication

Message authentication is a mechanism or service used to verify the integrity of a message. Message authentication assures that data received are exactly as sent (i.e., contain no modification, insertion, deletion, or replay). In many cases, there is

Message or data block M (variable length) P, L

P, L = padding plus length field

L bits

Hash value h(fixed length)

H

Figure 11.1 Cryptographic Hash Function; h = H(M)


a requirement that the authentication mechanism assures that purported identity of the sender is valid. When a hash function is used to provide message authentication, the hash function value is often referred to as a message digest.1

The essence of the use of a hash function for message authentication is as follows. The sender computes a hash value as a function of the bits in the message and transmits both the hash value and the message. The receiver performs the same hash calculation on the message bits and compares this value with the incoming hash value. If there is a mismatch, the receiver knows that the message (or possibly the hash value) has been altered (Figure 11.2a).

(b) Man-in-the-middle attack

Alice

Darth

Bob

BobAlice

COMPARE

data

data

data’

H

data

data’

data’

H

H

(a) Use of hash function to check data integrity

COMPARE

data

data

data’

HH

Figure 11.2 Attack Against Hash Function

1The topic of this section is invariably referred to as message authentication. However, the concepts and techniques apply equally to data at rest. For example, authentication techniques can be applied to a file in storage to assure that the file is not tampered with.


The hash function must be transmitted in a secure fashion. That is, the hash function must be protected so that if an adversary alters or replaces the message, it is not feasible for adversary to also alter the hash value to fool the receiver. This type of attack is shown in Figure 11.2b. In this example, Alice transmits a data block and attaches a hash value. Darth intercepts the message, alters or replaces the data block, and calculates and attaches a new hash value. Bob receives the altered data with the new hash value and does not detect the change. To prevent this attack, the hash value generated by Alice must be protected.

Figure 11.3 illustrates a variety of ways in which a hash code can be used to provide message authentication, as follows.

E

K

M

H

| | D

K

M

H(M)

H

Compare

(a)

M

H

| |

K

(b)

M

D

H

CompareK

E

E(K, [M || H(M)])

E(K, H(M))

Destination BSource A

| |SM

H

| |

S(c)

| |

M

H(M || S)

H(M || S)

H

Compare

M

H

| |

S(d)

| |

E

K

| |S H

Compare

MD

KE(K, [M || H(M || S)])

Figure 11.3 Simplified Examples of the Use of a Hash Function for Message Authentication


a. The message plus concatenated hash code is encrypted using symmetric encryption. Because only A and B share the secret key, the message must have come from A and has not been altered. The hash code provides the struc-ture or redundancy required to achieve authentication. Because encryption is applied to the entire message plus hash code, confidentiality is also provided.

b. Only the hash code is encrypted, using symmetric encryption. This reduces the processing burden for those applications that do not require confidentiality.

c. It is possible to use a hash function but no encryption for message authentication. The technique assumes that the two communicating parties share a common se-cret value S. A computes the hash value over the concatenation of M and S and appends the resulting hash value to M. Because B possesses S, it can recompute the hash value to verify. Because the secret value itself is not sent, an opponent cannot modify an intercepted message and cannot generate a false message.

d. Confidentiality can be added to the approach of method (c) by encrypting the entire message plus the hash code.

When confidentiality is not required, method (b) has an advantage over methods (a) and (d), which encrypts the entire message, in that less computa-tion is required. Nevertheless, there has been growing interest in techniques that avoid encryption (Figure 11.3c). Several reasons for this interest are pointed out in [TSUD92].

• Encryption software is relatively slow. Even though the amount of data to be encrypted per message is small, there may be a steady stream of messages into and out of a system.

• Encryption hardware costs are not negligible. Low-cost chip implementations of DES are available, but the cost adds up if all nodes in a network must have this capability.

• Encryption hardware is optimized toward large data sizes. For small blocks of data, a high proportion of the time is spent in initialization/invocation overhead.

• Encryption algorithms may be covered by patents, and there is a cost associ-ated with licensing their use.

More commonly, message authentication is achieved using a message authen-tication code (MAC), also known as a keyed hash function. Typically, MACs are used between two parties that share a secret key to authenticate information ex-changed between those parties. A MAC function takes as input a secret key and a data block and produces a hash value, referred to as the MAC, which is associated with the protected message. If the integrity of the message needs to be checked, the MAC function can be applied to the message and the result compared with the as-sociated MAC value. An attacker who alters the message will be unable to alter the associated MAC value without knowledge of the secret key. Note that the verifying party also knows who the sending party is because no one else knows the secret key.

Note that the combination of hashing and encryption results in an overall function that is, in fact, a MAC (Figure 11.3b). That is, E(K, H(M)) is a function of a variable-length message M and a secret key K, and it produces a fixed-size output


that is secure against an opponent who does not know the secret key. In practice, specific MAC algorithms are designed that are generally more efficient than an en-cryption algorithm.

We discuss MACs in Chapter 12.

Digital Signatures

Another important application, which is similar to the message authentication application, is the digital signature. The operation of the digital signature is similar to that of the MAC. In the case of the digital signature, the hash value of a message is encrypted with a user’s private key. Anyone who knows the user’s public key can verify the integrity of the message that is associated with the digital signature. In this case, an attacker who wishes to alter the message would need to know the user’s private key. As we shall see in Chapter 14, the implications of digital signa-tures go beyond just message authentication.

Figure 11.4 illustrates, in a simplified fashion, how a hash code is used to pro-vide a digital signature.

a. The hash code is encrypted, using public-key encryption with the sender’s pri-vate key. As with Figure 11.3b, this provides authentication. It also provides a digital signature, because only the sender could have produced the encrypted hash code. In fact, this is the essence of the digital signature technique.

b. If confidentiality as well as a digital signature is desired, then the message plus the private-key-encrypted hash code can be encrypted using a symmetric secret key. This is a common technique.

M

H

| | E

PRa PUa

E

K

D

K

M

D

H

Compare

(b)E(PRa, H(M))

E(K, [M || E(PRa, H(M))])


PRa PUa

M

H

| |

(a)

M

E D

H

Compare

E(PRa, H(M))

Figure 11.4 Simplified Examples of Digital Signatures


Other Applications

Hash functions are commonly used to create a one-way password file. Chapter 21 explains a scheme in which a hash of a password is stored by an operating system rather than the password itself. Thus, the actual password is not retrievable by a hacker who gains access to the password file. In simple terms, when a user en-ters a password, the hash of that password is compared to the stored hash value for verification. This approach to password protection is used by most operating systems.

Hash functions can be used for intrusion detection and virus detection. Store H(F) for each file on a system and secure the hash values (e.g., on a CD-R that is kept secure). One can later determine if a file has been modified by recomputing H(F). An intruder would need to change F without changing H(F).

A cryptographic hash function can be used to construct a pseudorandom function (PRF) or a pseudorandom number generator (PRNG). A common appli-cation for a hash-based PRF is for the generation of symmetric keys. We discuss this application in Chapter 12.

11.2 twO simpLe hash functiOns

To get some feel for the security considerations involved in cryptographic hash functions, we present two simple, insecure hash functions in this section. All hash functions operate using the following general principles. The input (message, file, etc.) is viewed as a sequence of n-bit blocks. The input is processed one block at a time in an iterative fashion to produce an n-bit hash function.

One of the simplest hash functions is the bit-by-bit exclusive-OR (XOR) of every block. This can be expressed as

Ci = bi1 ⊕ bi2 ⊕ g⊕ bim

where

Ci = ith bit of the hash code, 1 … i … n

m = number of n-bit blocks in the input

bij = ith bit in jth block⊕ = XOR operation

This operation produces a simple parity for each bit position and is known as a longitudinal redundancy check. It is reasonably effective for random data as a data integrity check. Each n-bit hash value is equally likely. Thus, the probability that a data error will result in an unchanged hash value is 2-n. With more predictably for-matted data, the function is less effective. For example, in most normal text files, the high-order bit of each octet is always zero. So if a 128-bit hash value is used, instead of an effectiveness of 2-128, the hash function on this type of data has an effective-ness of 2-112.

11.2 / two simple hash FunCtions 321

A simple way to improve matters is to perform a one-bit circular shift, or rotation, on the hash value after each block is processed. The procedure can be summarized as follows.

1. Initially set the n-bit hash value to zero.

2. Process each successive n-bit block of data as follows:

a. Rotate the current hash value to the left by one bit.

b. XOR the block into the hash value.

This has the effect of “randomizing” the input more completely and overcoming any regularities that appear in the input. Figure 11.5 illustrates these two types of hash functions for 16-bit hash values.

Although the second procedure provides a good measure of data integrity, it is virtually useless for data security when an encrypted hash code is used with a

XOR of every 16-bit blockXOR with 1-bit rotation to the right

16 bits

Figure 11.5 Two Simple Hash Functions


plaintext message, as in Figures 11.3b and 11.4a. Given a message, it is an easy mat-ter to produce a new message that yields that hash code: Simply prepare the desired alternate message and then append an n-bit block that forces the new message plus block to yield the desired hash code.

Although a simple XOR or rotated XOR (RXOR) is insufficient if only the hash code is encrypted, you may still feel that such a simple function could be use-ful when the message together with the hash code is encrypted (Figure 11.3a). But you must be careful. A technique originally proposed by the National Bureau of Standards used the simple XOR applied to 64-bit blocks of the message and then an encryption of the entire message that used the cipher block chaining (CBC) mode. We can define the scheme as follows: Given a message M consisting of a sequence of 64-bit blocks X1, X2, c, XN, define the hash code h = H(M) as the block-by-block XOR of all blocks and append the hash code as the final block:

h = XN+1 = X1 ⊕ X2 ⊕ g ⊕ XN

Next, encrypt the entire message plus hash code using CBC mode to produce the encrypted message Y1, Y2, c, YN+1. [JUEN85] points out several ways in which the ciphertext of this message can be manipulated in such a way that it is not detectable by the hash code. For example, by the definition of CBC (Figure 6.4), we have

X1 = IV ⊕ D(K, Y1)

Xi = Yi- 1 ⊕ D(K, Yi)

XN+1 = YN ⊕ D(K, YN+1)

But XN+1 is the hash code:

XN+1 = X1 ⊕ X2 ⊕ c ⊕ XN

= [IV ⊕ D(K, Y1)] ⊕ [Y1 ⊕ D(K, Y2)] ⊕ c⊕ [YN - 1 ⊕ D(K, YN)]

Because the terms in the preceding equation can be XORed in any order, it follows that the hash code would not change if the ciphertext blocks were permuted.

11.3 requirements and security

Before proceeding, we need to define two terms. For a hash value h = H(x), we say that x is the preimage of h. That is, x is a data block whose hash function, using the function H, is h. Because H is a many-to-one mapping, for any given hash value h, there will in general be multiple preimages. A collision occurs if we have x ≠ y and H(x) = H(y). Because we are using hash functions for data integrity, collisions are clearly undesirable.

Let us consider how many preimages are there for a given hash value, which is a measure of the number of potential collisions for a given hash value. Suppose the length of the hash code is n bits, and the function H takes as input messages or data blocks of length b bits with b 7 n. Then, the total number of possible messages is 2b and the total number of possible hash values is 2n. On average, each hash value corresponds to 2b - n preimages. If H tends to uniformly distribute hash values then, in fact, each hash value will have close to 2b - n preimages. If we now allow inputs of

11.3 / requirements and seCurity 323

arbitrary length, not just a fixed length of some number of bits, then the number of preimages per hash value is arbitrarily large. However, the security risks in the use of a hash function are not as severe as they might appear from this analysis. To un-derstand better the security implications of cryptographic hash functions, we need precisely define their security requirements.

Security Requirements for Cryptographic Hash Functions

Table 11.1 lists the generally accepted requirements for a cryptographic hash func-tion. The first three properties are requirements for the practical application of a hash function.

The fourth property, preimage resistant, is the one-way property: it is easy to generate a code given a message, but virtually impossible to generate a message given a code. This property is important if the authentication technique involves the use of a secret value (Figure 11.3c). The secret value itself is not sent. However, if the hash function is not one way, an attacker can easily discover the secret value: If the attacker can observe or intercept a transmission, the attacker obtains the message M, and the hash code h = H(S ‘ M). The attacker then inverts the hash function to obtain S ‘ M = H-1(MDM). Because the attacker now has both M and SAB ‘ M, it is a trivial matter to recover SAB.

The fifth property, second preimage resistant, guarantees that it is impossible to find an alternative message with the same hash value as a given message. This prevents forgery when an encrypted hash code is used (Figures 11.3b and 11.4a). If this property were not true, an attacker would be capable of the following sequence: First, observe or intercept a message plus its encrypted hash code; second, generate an unencrypted hash code from the message; third, generate an alternate message with the same hash code.

A hash function that satisfies the first five properties in Table 11.1 is referred to as a weak hash function. If the sixth property, collision resistant, is also satis-fied, then it is referred to as a strong hash function. A strong hash function protects

Table 11.1 Requirements for a Cryptographic Hash Function H

Requirement Description

Variable input size H can be applied to a block of data of any size.

Fixed output size H produces a fixed-length output.

Efficiency H(x) is relatively easy to compute for any given x, making both hardware and software implementations practical.

Preimage resistant (one-way property) For any given hash value h, it is computationally infeasible to find y such that H(y) = h.

Second preimage resistant (weak collision resistant)

For any given block x, it is computationally infeasible to find y ≠ x with H(y) = H(x).

Collision resistant (strong collision resistant)

It is computationally infeasible to find any pair (x, y) such that H(x) = H(y).

Pseudorandomness Output of H meets standard tests for pseudorandomness.


against an attack in which one party generates a message for another party to sign. For example, suppose Bob writes an IOU message, sends it to Alice, and she signs it. Bob finds two messages with the same hash, one of which requires Alice to pay a small amount and one that requires a large payment. Alice signs the first message, and Bob is then able to claim that the second message is authentic.

Figure 11.6 shows the relationships among the three resistant properties. A function that is collision resistant is also second preimage resistant, but the reverse is not necessarily true. A function can be collision resistant but not preim-age resistant and vice versa. A function can be preimage resistant but not second preimage resistant and vice versa. See [MENE97] for a discussion.

Table 11.2 shows the resistant properties required for various hash function applications.

The final requirement in Table 11.1, pseudorandomness, has not traditionally been listed as a requirement of cryptographic hash functions but is more or less im-plied. [JOHN05] points out that cryptographic hash functions are commonly used for key derivation and pseudorandom number generation, and that in message in-tegrity applications, the three resistant properties depend on the output of the hash function appearing to be random. Thus, it makes sense to verify that in fact a given hash function produces pseudorandom output.

Secondpreimage resistant

Preimageresistant

Collisionresistant

Figure 11.6 Relationship Among Hash Function Properties

Table 11.2 Hash Function Resistance Properties Required for Various Data Integrity Applications

Preimage Resistant Second Preimage Resistant

Collision Resistant

Hash + digital signature yes yes yes*

Intrusion detection and virus detection

yes

Hash + symmetric encryption

One-way password file yes

MAC yes yes yes*

*Resistance required if attacker is able to mount a chosen message attack


Brute-Force Attacks

As with encryption algorithms, there are two categories of attacks on hash functions: brute-force attacks and cryptanalysis. A brute-force attack does not de-pend on the specific algorithm but depends only on bit length. In the case of a hash function, a brute-force attack depends only on the bit length of the hash value. A cryptanalysis, in contrast, is an attack based on weaknesses in a particular crypto-graphic algorithm. We look first at brute-force attacks.

Preimage and Second Preimage attackS For a preimage or second preimage attack, an adversary wishes to find a value y such that H(y) is equal to a given hash value h. The brute-force method is to pick values of y at random and try each value until a collision occurs. For an m-bit hash value, the level of effort is proportional to 2m. Specifically, the adversary would have to try, on average, 2m - 1 values of y to find one that generates a given hash value h. This result is derived in Appendix 11A [Equation (11.1)].

colliSion reSiStant attackS For a collision resistant attack, an adversary wishes to find two messages or data blocks, x and y, that yield the same hash function: H(x) = H(y). This turns out to require considerably less effort than a preimage or second preimage attack. The effort required is explained by a mathematical result referred to as the birthday paradox. In essence, if we choose random variables from a uniform distribution in the range 0 through N - 1, then the probability that a repeated element is encountered exceeds 0.5 after 1N choices have been made. Thus, for an m-bit hash value, if we pick data blocks at random, we can expect to find two data blocks with the same hash value within 22m = 2m/2 attempts. The math-ematical derivation of this result is found in Appendix 11A.

Yuval proposed the following strategy to exploit the birthday paradox in a col-lision resistant attack [YUVA79].

1. The source, A, is prepared to sign a legitimate message x by appending the ap-propriate m-bit hash code and encrypting that hash code with A’s private key (Figure 11.4a).

2. The opponent generates 2m/2 variations x′ of x, all of which convey essentially the same meaning, and stores the messages and their hash values.

3. The opponent prepares a fraudulent message y for which A’s signature is desired.

4. The opponent generates minor variations y′ of y, all of which convey essen-tially the same meaning. For each y′, the opponent computes H(y′), checks for matches with any of the H(x′) values, and continues until a match is found. That is, the process continues until a y′ is generated with a hash value equal to the hash value of one of the x′ values.

5. The opponent offers the valid variation to A for signature. This signature can then be attached to the fraudulent variation for transmission to the intended recipient. Because the two variations have the same hash code, they will pro-duce the same signature; the opponent is assured of success even though the encryption key is not known.

Thus, if a 64-bit hash code is used, the level of effort required is only on the order of 232 [see Appendix 11A, Equation (11.7)].


The generation of many variations that convey the same meaning is not diffi-cult. For example, the opponent could insert a number of “space-space- backspace” character pairs between words throughout the document. Variations could then be generated by substituting “space-backspace-space” in selected instances. Alternatively, the opponent could simply reword the message but retain the mean-ing. Figure 11.7 provides an example.

To summarize, for a hash code of length m, the level of effort required, as we have seen, is proportional to the following.

Preimage resistant 2m

Second preimage resistant 2m

Collision resistant 2m/2

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If collision resistance is required (and this is desirable for a general-purpose secure hash code), then the value 2m/2 determines the strength of the hash code against brute-force attacks. Van Oorschot and Wiener [VANO94] presented a de-sign for a $10 million collision search machine for MD5, which has a 128-bit hash length, that could find a collision in 24 days. Thus, a 128-bit code may be viewed as inadequate. The next step up, if a hash code is treated as a sequence of 32 bits, is a 160-bit hash length. With a hash length of 160 bits, the same search machine would require over four thousand years to find a collision. With today’s technology, the time would be much shorter, so that 160 bits now appears suspect.

Cryptanalysis

As with encryption algorithms, cryptanalytic attacks on hash functions seek to exploit some property of the algorithm to perform some attack other than an exhaustive search. The way to measure the resistance of a hash algorithm to cryptanalysis is to compare its strength to the effort required for a brute-force attack. That is, an ideal hash algorithm will require a cryptanalytic effort greater than or equal to the brute-force effort.

In recent years, there has been considerable effort, and some successes, in developing cryptanalytic attacks on hash functions. To understand these, we need to look at the overall structure of a typical secure hash function, indicated in Figure 11.8. This structure, referred to as an iterated hash function, was proposed by Merkle [MERK79, MERK89] and is the structure of most hash functions in use today, including SHA, which is discussed later in this chapter. The hash function takes an input message and partitions it into L fixed-sized blocks of b bits each. If necessary, the final block is padded to b bits. The final block also includes the value of the total length of the input to the hash function. The inclusion of the length makes the job of the opponent more difficult. Either the opponent must find two messages of equal length that hash to the same value or two messages of differing lengths that, together with their length values, hash to the same value.

The hash algorithm involves repeated use of a compression function, f, that takes two inputs (an n-bit input from the previous step, called the chaining variable, and a b-bit block) and produces an n-bit output. At the start of hashing, the chain-ing variable has an initial value that is specified as part of the algorithm. The final

f fn nn

IV �CV0 CV1

b

n

CVL�1

CVLn

b

Y0 Y1 YL�1

IV � Initial valueCVi � Chaining variableYi � ith input blockf � Compression algorithm

L � Number of input blocksn � Length of hash codeb � Length of input block

b

f

Figure 11.8 General Structure of Secure Hash Code


value of the chaining variable is the hash value. Often, b 7 n; hence the term com-pression. The hash function can be summarized as

CV0 = IV = initial n@bit value

CVi = f(CVi- 1, Yi- 1) 1 … i … L

H(M) = CVL

where the input to the hash function is a message M consisting of the blocks Y0, Y1, c, YL - 1.

The motivation for this iterative structure stems from the observation by Merkle [MERK89] and Damgard [DAMG89] that if the compression function is col-lision resistant, then so is the resultant iterated hash function.2 Therefore, the struc-ture can be used to produce a secure hash function to operate on a message of any length. The problem of designing a secure hash function reduces to that of designing a collision-resistant compression function that operates on inputs of some fixed size.

Cryptanalysis of hash functions focuses on the internal structure of f and is based on attempts to find efficient techniques for producing collisions for a single execution of f. Once that is done, the attack must take into account the fixed value of IV. The attack on f depends on exploiting its internal structure. Typically, as with symmetric block ciphers, f consists of a series of rounds of processing, so that the attack involves analysis of the pattern of bit changes from round to round.

Keep in mind that for any hash function there must exist collisions, because we are mapping a message of length at least equal to twice the block size b (because we must append a length field) into a hash code of length n, where b Ú n. What is required is that it is computationally infeasible to find collisions.

The attacks that have been mounted on hash functions are rather complex and beyond our scope here. For the interested reader, [DOBB96] and [BELL97] are recommended.

11.4 hash functiOns based On cipher bLOck chaining

A number of proposals have been made for hash functions based on using a cipher block chaining technique, but without using the secret key. One of the first such proposals was that of Rabin [RABI78]. Divide a message M into fixed-size blocks M1, M2, c , MN and use a symmetric encryption system such as DES to compute the hash code G as

H0 = initial value

Hi = E(Mi, Hi- 1)

G = HN

This is similar to the CBC technique, but in this case, there is no secret key. As with any hash code, this scheme is subject to the birthday attack, and if the encryption al-gorithm is DES and only a 64-bit hash code is produced, then the system is vulnerable.

2The converse is not necessarily true.

11.5 / seCure hash algorithm (sha) 329

Furthermore, another version of the birthday attack can be used even if the opponent has access to only one message and its valid signature and cannot obtain multiple signings. Here is the scenario: We assume that the opponent intercepts a message with a signature in the form of an encrypted hash code and that the unen-crypted hash code is m bits long.

1. Use the algorithm defined at the beginning of this subsection to calculate the unencrypted hash code G.

2. Construct any desired message in the form Q1, Q2, c, QN - 2.

3. Compute Hi = E(Qi, Hi- 1) for 1 … i … (N - 2).

4. Generate 2m/2 random blocks; for each block X, compute E(X, HN - 2). Generate an additional 2m/2 random blocks; for each block Y , compute D(Y , G), where D is the decryption function corresponding to E.

5. Based on the birthday paradox, with high probability there will be an X and Y such that E(X, HN - 2) = D(Y, G).

6. Form the message Q1, Q2, c , QN - 2, X, Y. This message has the hash code G and therefore can be used with the intercepted encrypted signature.

This form of attack is known as a meet-in-the-middle-attack. A number of researchers have proposed refinements intended to strengthen the basic block chaining approach. For example, Davies and Price [DAVI89] describe the variation:

Hi = E(Mi, Hi- 1) ⊕ Hi- 1

Another variation, proposed in [MEYE88], is

Hi = E(Hi- 1, Mi) ⊕ Mi

However, both of these schemes have been shown to be vulnerable to a vari-ety of attacks [MIYA90]. More generally, it can be shown that some form of birth-day attack will succeed against any hash scheme involving the use of cipher block chaining without a secret key, provided that either the resulting hash code is small enough (e.g., 64 bits or less) or that a larger hash code can be decomposed into independent subcodes [JUEN87].

Thus, attention has been directed at finding other approaches to hashing. Many of these have also been shown to have weaknesses [MITC92].

11.5 secure hash aLgOrithm (sha)

In recent years, the most widely used hash function has been the Secure Hash Algorithm (SHA). Indeed, because virtually every other widely used hash function had been found to have substantial cryptanalytic weaknesses, SHA was more or less the last remaining standardized hash algorithm by 2005. SHA was developed by the National Institute of Standards and Technology (NIST) and published as a federal information processing standard (FIPS 180) in 1993. When weaknesses were discovered in SHA, now known as SHA-0, a revised version was issued as FIPS 180-1 in 1995 and is referred to as SHA-1. The actual standards document is entitled


“Secure Hash Standard.” SHA is based on the hash function MD4, and its design closely models MD4.

SHA-1 produces a hash value of 160 bits. In 2002, NIST produced a revised version of the standard, FIPS 180-2, that defined three new versions of SHA, with hash value lengths of 256, 384, and 512 bits, known as SHA-256, SHA-384, and SHA-512, respectively. Collectively, these hash algorithms are known as SHA-2. These new versions have the same underlying structure and use the same types of modular arithmetic and logical binary operations as SHA-1. A revised document was issued as FIP PUB 180-3 in 2008, which added a 224-bit version (Table 11.3). SHA-1 and SHA-2 are also specified in RFC 6234, which essentially duplicates the material in FIPS 180-3 but adds a C code implementation.

In 2005, NIST announced the intention to phase out approval of SHA-1 and move to a reliance on SHA-2 by 2010. Shortly thereafter, a research team de-scribed an attack in which two separate messages could be found that deliver the same SHA-1 hash using 269 operations, far fewer than the 280 operations previously thought needed to find a collision with an SHA-1 hash [WANG05]. This result should hasten the transition to SHA-2.

In this section, we provide a description of SHA-512. The other versions are quite similar.

SHA-512 Logic

The algorithm takes as input a message with a maximum length of less than 2128 bits and produces as output a 512-bit message digest. The input is processed in 1024-bit blocks. Figure 11.9 depicts the overall processing of a message to produce a digest. This follows the general structure depicted in Figure 11.8. The processing consists of the following steps.

Step 1 Append padding bits. The message is padded so that its length is congruent to 896 modulo 1024 [length K 896(mod 1024)]. Padding is always added, even if the message is already of the desired length. Thus, the number of padding bits is in the range of 1 to 1024. The padding consists of a single 1 bit followed by the necessary number of 0 bits.

Step 2 Append length. A block of 128 bits is appended to the message. This block is treated as an unsigned 128-bit integer (most significant byte first) and contains the length of the original message (before the padding).

Table 11.3 Comparison of SHA Parameters

SHA-1 SHA-224 SHA-256 SHA-384 SHA-512

Message Digest Size 160 224 256 384 512

Message Size 6 264 6 264 6 264 6 2128 6 2128

Block Size 512 512 512 1024 1024

Word Size 32 32 32 64 64

Number of Steps 80 64 64 80 80

Note: All sizes are measured in bits.


The outcome of the first two steps yields a message that is an inte-ger multiple of 1024 bits in length. In Figure 11.9, the expanded message is represented as the sequence of 1024-bit blocks M1, M2, c, MN, so that the total length of the expanded message is N * 1024 bits.

Step 3 Initialize hash buffer. A 512-bit buffer is used to hold intermediate and final results of the hash function. The buffer can be represented as eight 64-bit registers (a, b, c, d, e, f, g, h). These registers are initialized to the following 64-bit integers (hexadecimal values):

a = 6A09E667F3BCC908 e = 510E527FADE682D1

b = BB67AE8584CAA73B f = 9B05688C2B3E6C1F

c = 3C6EF372FE94F82B g = 1F83D9ABFB41BD6B

d = A54FF53A5F1D36F1 h = 5BE0CD19137E2179

These values are stored in big-endian format, which is the most significant byte of a word in the low-address (leftmost) byte position. These words were obtained by taking the first sixty-four bits of the fractional parts of the square roots of the first eight prime numbers.

Step 4 Process message in 1024-bit (128-word) blocks. The heart of the algorithm is a module that consists of 80 rounds; this module is labeled F in Figure 11.9. The logic is illustrated in Figure 11.10.

N 1024 bits

M1

H1

M2 MN

F

IV = H0

Message

hash code

1024 bits

512 bits 512 bits 512 bits

1024 bits 1024 bits

L bits

L

128 bits

1000000, . . . ,0

+

H2

F

+

HN

F

+

+ = word-by-word addition mod 264

Figure 11.9 Message Digest Generation Using SHA-512


Each round takes as input the 512-bit buffer value, abcdefgh, and up-dates the contents of the buffer. At input to the first round, the buffer has the value of the intermediate hash value, Hi- 1. Each round t makes use of a 64-bit value Wt, derived from the current 1024-bit block being processed (Mi). These values are derived using a message schedule described sub-sequently. Each round also makes use of an additive constant Kt, where 0 … t … 79 indicates one of the 80 rounds. These words represent the first 64 bits of the fractional parts of the cube roots of the first 80 prime numbers. The constants provide a “randomized” set of 64-bit patterns, which should eliminate any regularities in the input data. Table 11.4 shows these constants in hexadecimal format (from left to right).

The output of the eightieth round is added to the input to the first round (Hi- 1) to produce Hi. The addition is done independently for each of the eight words in the buffer with each of the corresponding words in Hi- 1, using addition modulo 264.

Step 5 Output. After all N 1024-bit blocks have been processed, the output from the Nth stage is the 512-bit message digest.

64

Mi

Wt

Hi

Hi�1

W0

W79

Kt

K0

K79

a b c

Round 0

d e f g h

a b c

Round t

d e f g h

Messageschedule

a b c

Round 79

d e f g h

� � � � � � � �

Figure 11.10 SHA-512 Processing of a Single 1024-Bit Block


We can summarize the behavior of SHA-512 as follows:

H0 = IV

Hi = SUM64(Hi- 1, abcdefghi)

MD = HN

where

IV = initial value of the abcdefgh buffer, defined in step 3

abcdefghi = the output of the last round of processing of the ith message block

N = the number of blocks in the message (including padding and length fields)

SUM64 = addition modulo 264 performed separately on each word of the pair of inputs

MD = final message digest value

Table 11.4 SHA-512 Constants

428a2f98d728ae22 7137449123ef65cd b5c0fbcfec4d3b2f e9b5dba58189dbbc

3956c25bf348b538 59f111f1b605d019 923f82a4af194f9b ab1c5ed5da6d8118

d807aa98a3030242 12835b0145706fbe 243185be4ee4b28c 550c7dc3d5ffb4e2

72be5d74f27b896f 80deb1fe3b1696b1 9bdc06a725c71235 c19bf174cf692694

e49b69c19ef14ad2 efbe4786384f25e3 0fc19dc68b8cd5b5 240ca1cc77ac9c65

2de92c6f592b0275 4a7484aa6ea6e483 5cb0a9dcbd41fbd4 76f988da831153b5

983e5152ee66dfab a831c66d2db43210 b00327c898fb213f bf597fc7beef0ee4

c6e00bf33da88fc2 d5a79147930aa725 06ca6351e003826f 142929670a0e6e70

27b70a8546d22ffc 2e1b21385c26c926 4d2c6dfc5ac42aed 53380d139d95b3df

650a73548baf63de 766a0abb3c77b2a8 81c2c92e47edaee6 92722c851482353b

a2bfe8a14cf10364 a81a664bbc423001 c24b8b70d0f89791 c76c51a30654be30

d192e819d6ef5218 d69906245565a910 f40e35855771202a 106aa07032bbd1b8

19a4c116b8d2d0c8 1e376c085141ab53 2748774cdf8eeb99 34b0bcb5e19b48a8

391c0cb3c5c95a63 4ed8aa4ae3418acb 5b9cca4f7763e373 682e6ff3d6b2b8a3

748f82ee5defb2fc 78a5636f43172f60 84c87814a1f0ab72 8cc702081a6439ec

90befffa23631e28 a4506cebde82bde9 bef9a3f7b2c67915 c67178f2e372532b

ca273eceea26619c d186b8c721c0c207 eada7dd6cde0eb1e f57d4f7fee6ed178

06f067aa72176fba 0a637dc5a2c898a6 113f9804bef90dae 1b710b35131c471b

28db77f523047d84 32caab7b40c72493 3c9ebe0a15c9bebc 431d67c49c100d4c

4cc5d4becb3e42b6 597f299cfc657e2a 5fcb6fab3ad6faec 6c44198c4a475817


SHA-512 Round Function

Let us look in more detail at the logic in each of the 80 steps of the processing of one 512-bit block (Figure 11.11). Each round is defined by the following set of equations:

T1 = h + Ch(e, f, g) + 1a 5121 e2 + Wt + Kt

T2 = 1a 5120 a2 + Maj(a, b, c)

h = g

g = f

f = e

e = d + T1

d = c

c = b

b = a

a = T1 + T2

where

t = step number; 0 … t … 79

Ch(e, f, g) = (e AND f) ⊕ (NOT e AND g)the conditional function: If e then f else g

Maj(a, b, c) = (a AND b) ⊕ (a AND c) ⊕ (b AND c) the function is true only of the majority (two or three) of the arguments are true

1a 5120 a2 = ROTR28(a) ⊕ ROTR34(a) ⊕ ROTR39(a)

1a 5121 e2 = ROTR14(e) ⊕ ROTR18(e) ⊕ ROTR41(e)

ROTRn(x) = circular right shift (rotation) of the 64-bit argument x by n bits

a b c d e f g h

a b c d512 bits

e f g h

Ch

Kt

Wt

Maj

+

+

+

+

+

+

+

Figure 11.11 Elementary SHA-512 Operation (single round)


Wt = a 64-bit word derived from the current 1024-bit input block

Kt = a 64-bit additive constant

+ = addition modulo 264

Two observations can be made about the round function.

1. Six of the eight words of the output of the round function involve simply per-mutation (b, c, d, f , g, h) by means of rotation. This is indicated by shading in Figure 11.11.

2. Only two of the output words (a, e) are generated by substitution. Word e is a function of input variables (d, e, f , g, h), as well as the round word Wt and the constant Kt. Word a is a function of all of the input variables except d, as well as the round word Wt and the constant Kt.

It remains to indicate how the 64-bit word values Wt are derived from the 1024-bit message. Figure 11.12 illustrates the mapping. The first 16 values of Wt are taken directly from the 16 words of the current block. The remaining values are defined as

Wt = s1512(Wt- 2) + Wt- 7 + s0

512(Wt- 15) + Wt- 16

where

s0512(x) = ROTR1(x) ⊕ ROTR8(x) ⊕ SHR7(x)

s1512(x) = ROTR19(x) ⊕ ROTR61(x) ⊕ SHR6(x)

ROTRn(x) = circular right shift (rotation) of the 64-bit argument x by n bits

SHRn(x) = left shift of the 64-bit argument x by n bits with padding by zeros on the right

+ = addition modulo 264

Thus, in the first 16 steps of processing, the value of Wt is equal to the cor-responding word in the message block. For the remaining 64 steps, the value of Wt consists of the circular left shift by one bit of the XOR of four of the preced-ing values of Wt, with two of those values subjected to shift and rotate operations. This introduces a great deal of redundancy and interdependence into the message

1024 bits

64 bits

Wt–16W0 W1 W9 W14 W63 W65 W71 W76Wt–15 Wt–7 Wt–2

W0 W1 W15 W16 Wt

Mi

W79

+

σ0 σ1 σ0 σ1 σ0 σ1

+ +

Figure 11.12 Creation of 80-word Input Sequence for SHA-512 Processing of Single Block


blocks that are compressed, which complicates the task of finding a different mes-sage block that maps to the same compression function output.

Figure 11.13 summarizes the SHA-512 logic.The SHA-512 algorithm has the property that every bit of the hash code is a

function of every bit of the input. The complex repetition of the basic function F produces results that are well mixed; that is, it is unlikely that two messages chosen at random, even if they exhibit similar regularities, will have the same hash code. Unless there is some hidden weakness in SHA-512, which has not so far been pub-lished, the difficulty of coming up with two messages having the same message di-gest is on the order of 2256 operations, while the difficulty of finding a message with a given digest is on the order of 2512 operations.

Example

We include here an example based on one in FIPS 180. We wish to hash a one-block message consisting of three ASCII characters: “abc,” which is equivalent to the fol-lowing 24-bit binary string:

01100001 01100010 01100011

Recall from step 1 of the SHA algorithm, that the message is padded to a length congruent to 896 modulo 1024. In this case of a single block, the padding consists of 896 - 24 = 872 bits, consisting of a “1” bit followed by 871 “0” bits. Then a 128-bit length value is appended to the message, which contains the length of the original message (before the padding). The original length is 24 bits, or a hexadecimal value of 18. Putting this all together, the 1024-bit message block, in hexadecimal, is

6162638000000000 0000000000000000 0000000000000000 0000000000000000

0000000000000000 0000000000000000 0000000000000000 0000000000000000

0000000000000000 0000000000000000 0000000000000000 0000000000000000

0000000000000000 0000000000000000 0000000000000000 0000000000000018

This block is assigned to the words W0, c,W15 of the message schedule, which appears as follows.

W0 = 6162638000000000 W8 = 0000000000000000

W1 = 0000000000000000 W9 = 0000000000000000

W2 = 0000000000000000 W10 = 0000000000000000

W3 = 0000000000000000 W11 = 0000000000000000

W4 = 0000000000000000 W12 = 0000000000000000

W5 = 0000000000000000 W13 = 0000000000000000

W6 = 0000000000000000 W14 = 0000000000000000

W7 = 0000000000000000 W15 = 0000000000000018


The padded message consists blocks M1, M2, . . . , MN. Each message block Mi consists of 16 64-bit words Mi,0, Mi,1, . . . , Mi,15. All addition is performed modulo 264.

H0,0 = 6A09E667F3BCC908 H0,4 = 510E527FADE682D1

H0,1 = BB67AE8584CAA73B H0,5 = 9B05688C2B3E6C1F

H0,2 = 3C6EF372FE94F82B H0,6 = 1F83D9ABFB41BD6B

H0,3 = A54FF53A5F1D36F1 H0,7 = 5BE0CDI9137E2179

for i = 1 to N

1. Prepare the message schedule Wfor t = 0 to 15

Wt = Mi,tfor t = 16 to 79

Wt = s5121 (Wt- 2) + Wt- 7 + s512

0 (Wt- 15) + Wt- 16

2. Initialize the working variablesa = Hi- 1, 0 e = Hi- 1, 4b = Hi- 1, 1 f = Hi- 1, 5c = Hi- 1, 2 g = Hi- 1, 6d = Hi- 1, 3 h = Hi- 1, 7

3. Perform the main hash computationfor t = 0 to 79

T1 = h + Ch(e, f, g) + aa 5121 eb + Wt + Kt

T2 = aa 5121 ab + Maj(a, b, c)

h = gg = ff = ee = d + T1d = cc = bb = aa = T1 + T2

4. Compute the intermediate hash valueHi, 0 = a + Hi- 1, 0 Hi, 4 = e + Hi- 1,4Hi, 1 = b + Hi- 1, 1 Hi, 5 = f + Hi- 1, 5Hi, 2 = c + Hi- 1, 2 Hi, 6 = g + Hi- 1, 6Hi, 3 = d + Hi- 1, 3 Hi, 7 = h + Hi- 1, 7

return {HN, 0 }HN, 1 }HN, 2 }HN, 3 }HN, 4 }HN, 5 }HN, 6 }HN, 7}

Figure 11.13 SHA-512 Logic


As indicated in Figure 11.13, the eight 64-bit variables, a through h, are initial-ized to values H0,0 through H0,7. The following table shows the initial values of these variables and their values after each of the first two rounds.

a 6a09e667f3bcc908 f6afceb8bcfcddf5 1320f8c9fb872cc0

b bb67ae8584caa73b 6a09e667f3bcc908 f6afceb8bcfcddf5

c 3c6ef372fe94f82b bb67ae8584caa73b 6a09e667f3bcc908

d a54ff53a5f1d36f1 3c6ef372fe94f82b bb67ae8584caa73b

e 510e527fade682d1 58cb02347ab51f91 c3d4ebfd48650ffa

f 9b05688c2b3e6c1f 510e527fade682d1 58cb02347ab51f91

g 1f83d9abfb41bd6b 9b05688c2b3e6c1f 510e527fade682d1

h 5be0cd19137e2179 1f83d9abfb41bd6b 9b05688c2b3e6c1f

Note that in each of the rounds, six of the variables are copied directly from variables from the preceding round.

The process continues through 80 rounds. The output of the final round is

73a54f399fa4b1b2 10d9c4c4295599f6 d67806db8b148677 654ef9abec389ca9

d08446aa79693ed7 9bb4d39778c07f9e 25c96a7768fb2aa3 ceb9fc3691ce8326

The hash value is then calculated as

H1,0 = 6a09e667f3bcc908 + 73a54f399fa4b1b2 = ddaf35a193617aba

H1,1 = bb67ae8584caa73b + 10d9c4c4295599f6 = cc417349ae204131

H1,2 = 3c6ef372fe94f82b + d67806db8b148677 = 12e6fa4e89a97ea2

H1,3 = a54ff53a5f1d36f1 + 654ef9abec389ca9 = 0a9eeee64b55d39a

H1,4 = 510e527fade682d1 + d08446aa79693ed7 = 2192992a274fc1a8

H1,5 = 9b05688c2b3e6c1f + 9bb4d39778c07f9e = 36ba3c23a3feebbd

H1,6 = 1f83d9abfb41bd6b + 25c96a7768fb2aa3 = 454d4423643ce80e

H1,7 = 5be0cd19137e2179 + ceb9fc3691ce8326 = 2a9ac94fa54ca49f

ddaf35a193617aba cc417349ae204131 12e6fa4e89a97ea2 0a9eeee64b55d39a

2192992a274fc1a8 36ba3c23a3feebbd 454d4423643ce80e 2a9ac94fa54ca49f

The resulting 512-bit message digest is

Suppose now that we change the input message by one bit, from “abc” to “cbc.” Then, the 1024-bit message block is

6362638000000000 0000000000000000 0000000000000000 0000000000000000

0000000000000000 0000000000000000 0000000000000000 0000000000000000

0000000000000000 0000000000000000 0000000000000000 0000000000000000

0000000000000000 0000000000000000 0000000000000000 0000000000000018

11.6 / sha-3 339

The number of bit positions that differ between the two hash values is 253, almost exactly half the bit positions, indicating that SHA-512 has a good avalanche effect.

11.6 sha-3

As of this writing, the Secure Hash Algorithm (SHA-1) has not yet been “broken.” That is, no one has demonstrated a technique for producing collisions in a practical amount of time. However, because SHA-1 is very similar, in structure and in the basic mathematical operations used, to MD5 and SHA-0, both of which have been broken, SHA-1 is considered insecure and has been phased out for SHA-2.

SHA-2, particularly the 512-bit version, would appear to provide unassailable security. However, SHA-2 shares the same structure and mathematical operations as its predecessors, and this is a cause for concern. Because it will take years to find a suitable replacement for SHA-2, should it become vulnerable, NIST decided to begin the process of developing a new hash standard.

Accordingly, NIST announced in 2007 a competition to produce the next gen-eration NIST hash function, to be called SHA-3. The winning design for SHA-3 was announced by NIST in October 2012. SHA-3 is a cryptographic hash function that is intended to complement SHA-2 as the approved standard for a wide range of applications.

Appendix V looks at the evaluation criteria used by NIST to select from among the candidates for AES, plus the rationale for picking Keccak, which was the winning candidate. This material is useful in understanding not just the SHA-3 design but also the criteria by which to judge any cryptographic hash algorithm.

The Sponge Construction

The underlying structure of SHA-3 is a scheme referred to by its designers as a sponge construction [BERT07, BERT11]. The sponge construction has the same general structure as other iterated hash functions (Figure 11.8). The sponge func-tion takes an input message and partitions it into fixed-size blocks. Each block is processed in turn with the output of each iteration fed into the next iteration, finally producing an output block.

The sponge function is defined by three parameters:

f = the internal function used to process each input block3

r = the size in bits of the input blocks, called the bitrate pad = the padding algorithm

And the resulting 512-bit message digest is

531668966ee79b70 0b8e593261101354 4273f7ef7b31f279 2a7ef68d53f93264

319c165ad96d9187 55e6a204c2607e27 6e05cdf993a64c85 ef9e1e125c0f925f

3The Keccak documentation refers to f as a permutation. As we shall see, it involves both permutations and substitutions. We refer to f as the iteration function, because it is the function that is executed once for each iteration, that is, once for each block of the message that is processed.


A sponge function allows both variable length input and output, making it a flexible structure that can be used for a hash function (fixed-length output), a pseu-dorandom number generator (fixed-length input), and other cryptographic func-tions. Figure 11.14 illustrates this point. An input message of n bits is partitioned into k fixed-size blocks of r bits each. If necessary, the message is padded to achieve a length that is an integer multiple of r bits. The resulting partition is the sequence of blocks P0, P1, . . . , Pk–1, with n = k * r. For uniformity, padding is always added, so that if n mod r = 0, a padding block of r bits is added. The actual padding algorithm is a parameter of the function. The sponge specification proposes [BERT11] pro-poses two padding schemes:

• Simple padding: Denoted by pad10*, appends a single bit 1 followed by the minimum number of bits 0 such that the length of the result is a multiple of the block length.

• Multirate padding: Denoted by pad10*1, appends a single bit 1 followed by the minimum number of bits 0 followed by a single bit 1 such that the length of the result is a multiple of the block length. This is the simplest padding scheme that allows secure use of the same f with different rates r.

After processing all of the blocks, the sponge function generates a sequence of output blocks Z0, Z1, . . . , Zj–1. The number of output blocks generated is determined by the number of output bits desired. If the desired output is ℓ bits, then j blocks are produced, such that ( j - 1) * r 6 ℓ … j * r.

k r bits

(a) Input

(b) Output

P0 P1

Z0 Z1

Zj–1

Pk–1

message pad

r bits r bits r bits

r bits r bits r bits

l bits

n bits

Figure 11.14 Sponge Function Input and Output

11.6 / sha-3 341

Figure 11.15 shows the iterated structure of the sponge function. The sponge construction operates on a state variable s of b = r + c bits, which is initialized to all zeros and modified at each iteration. The value r is called the bitrate. This value is the block size used to partition the input message. The term bitrate reflects the fact that r is the number of bits processed at each iteration: the larger the value of r, the greater the rate at which message bits are processed by the sponge construction. The value c is referred to as the capacity. A discussion of the security implications of the capacity is beyond our scope. In essence, the capacity is a measure of the achiev-able complexity of the sponge construction and therefore the achievable level of security. A given implementation can trade claimed security for speed by increasing

(a) Absorbing phase

(b) Squeezing phase

f

r c

0c

0c

0c

0r 0c

P0

P1

P2

f

s

f

s

f

s

0cPk–1

b

r c

b

r c

Z0

r

Z1

Figure 11.15 Sponge Construction


the capacity c and decreasing the bitrate r accordingly, or vice versa. The default values for Keccak are c = 1024 bits, r = 576 bits, and therefore b = 1600 bits.

The sponge construction consists of two phases. The absorbing phase proceeds as follows: For each iteration, the input block to be processed is padded with zeroes to extend its length from r bits to b bits. Then, the bitwise XOR of the extended message block and s is formed to create a b-bit input to the iteration function f. The output of f is the value of s for the next iteration.

If the desired output length ℓ satisfies ℓ … b, then at the completion of the absorbing phase, the first ℓ bits of s are returned and the sponge construction termi-nates. Otherwise, the sponge construction enters the squeezing phase. To begin, the first ℓ bits of s are retained as block Z0. Then, the value of s is updated with repeated executions of f, and at each iteration, the first ℓ bits of s are retained as block Zi and concatenated with previously generated blocks. The process continues through ( j - 1) iterations until we have ( j - 1) * r 6 ℓ … j * r. At this point the first ℓ bits of the concatenated block Y are returned.

Note that the absorbing phase has the structure of a typical hash function. A common case will be one in which the desired hash length is less than or equal to the input block length; that is, ℓ … r. In that case, the sponge construction ter-minates after the absorbing phase. If a longer output than b bits is required, then the squeezing phase is employed. Thus the sponge construction is quite flexible. For example, a short message with a length r could be used as a seed and the sponge construction would function as a pseudorandom number generator.

To summarize, the sponge construction is a simple iterated construction for building a function F with variable-length input and arbitrary output length based on a fixed-length transformation or permutation f operating on a fixed number b of bits. The sponge construction is defined formally in [BERT11] as follows:

Algorithm The sponge construction SPONGE[f, pad, r]

Require: r 6 b

Interface: Z = sponge(M, /) with M ∈ Z2*, integer / 7 0 and Y ∈ Z2

/

P = M } pad[r](|M|) s = 0b

for i = 0 to |P|r − 1 do

s = s ⊕ (Pi } 0b - r) s = f(s)

end for

Z = :s;r while |Z|r r 6 / do s = f (s)

Z = Z } :s;r end while

return :Z; /

11.6 / sha-3 343

In the algorithm definition, the following notation is used: |M| is the length in bits of a bit string M. A bit string M can be considered as a sequence of blocks of some fixed length x, where the last block may be shorter. The number of blocks of M is denoted by |M|x. The blocks of M are denoted by Mi and the index ranges from 0 to |M|x - 1. The expression :M; / denotes the truncation of M to its first ℓ bits.

SHA-3 makes use of the iteration function f, labeled Keccak-f, which is de-scribed in the next section. The overall SHA-3 function is a sponge function expressed as Keccak[r, c] to reflect that SHA-3 has two operational parameters, r, the message block size, and c, the capacity, with the default of r + c = 1600 bits. Table 11.5 shows the supported values of r and c. As Table 11.5 shows, the hash function security as-sociated with the sponge construction is a function of the capacity c.

In terms of the sponge algorithm defined above, Keccak[r, c] is defined as

Keccak[r, c] ∆ SPONGE[Keccak@f [r + c], pad10 * 1, r]

We now turn to a discussion of the iteration function Keccak-f.

The SHA-3 Iteration Function f

We now examine the iteration function Keccak-f used to process each successive block of the input message. Recall that f takes as input a 1600-bit variable s consisting of r bits, corresponding to the message block size followed by c bits, referred to as the capacity. For internal processing within f, the input state variable s is organized as a 5 * 5 * 64 array a. The 64-bit units are referred to as lanes. For our purposes, we gen-erally use the notation a[x, y, z] to refer to an individual bit with the state array. When we are more concerned with operations that affect entire lanes, we designate the 5 * 5 matrix as L[x, y], where each entry in L is a 64-bit lane. The use of indices within this matrix is shown in Figure 11.16.4 Thus, the columns are labeled x = 0 through x = 4, the rows are labeled y = 0 through y = 4, and the individual bits

Table 11.5 SHA-3 Parameters

Message Digest Size 224 256 384 512

Message Size no maximum no maximum no maximum no maximum

Block Size (bitrate r) 1152 1088 832 576

Word Size 64 64 64 64

Number of Rounds 24 24 24 24

Capacity c 448 512 768 1024

Collision Resistance 2112 2128 2192 2256

Second Preimage Resistance

2224 2256 2384 2512

Note: All sizes and security levels—are measured in bits.

4Note that the first index (x) designates a column and the second index (y) designates a row. This is in conflict with the convention used in most mathematics sources, where the first index designates a row and the second index designates a column (e.g., Knuth, D. The Art of Computing Programming, Volume 1, Fundamental Algorithms; and Korn, G., and Korn, T. Mathematical Handbook for Scientists and Engineers).


within a lane are labeled z = 0 through z = 63. The mapping between the bits of s and those of a is

s[64(5y + x) + z] = a[x, y, z]

We can visualize this with respect to the matrix in Figure 11.16. When treating the state as a matrix of lanes, the first lane in the lower left corner, L[0, 0], corre-sponds to the first 64 bits of s. The lane in the second column, lowest row, L[1, 0], corresponds to the next 64 bits of s. Thus, the array a is filled with the bits of s start-ing with row y = 0 and proceeding row by row.

Structure of f The function f is executed once for each input block of the message to be hashed. The function takes as input the 1600-bit state variable and converts it into a 5 * 5 matrix of 64-bit lanes. This matrix then passes through 24 rounds of processing. Each round consists of five steps, and each step updates the state matrix by permutation or substitution operations. As shown in Figure 11.17, the rounds are identical with the exception of the final step in each round, which is modified by a round constant that differs for each round.

The application of the five steps can be expressed as the composition5 of functions:

R = i o x o p o r o u

Table 11.6 summarizes the operation of the five steps. The steps have a sim-ple description leading to a specification that is compact and in which no trapdoor can be hidden. The operations on lanes in the specification are limited to bitwise

L[0, 4]

x = 0 x = 1 x = 2 x = 3 x = 4

L[0, 3]

L[0, 2]

L[0, 1]

L[0, 0]

a[x, y, 0] a[x, y, 1] a[x, y, 2]

y = 1

y = 0

y = 2

y = 3

y = 4 L[1, 4]

L[1, 3]

L[1, 2]

L[1, 1]

L[1, 0]

L[2, 4]

L[2, 3]

L[2, 2]

L[2, 1]

L[2, 0]

(a) State variable as 5 5 matrix A of 64-bit words

(b) Bit labeling of 64-bit words

L[3, 4]

L[3, 3]

L[3, 2]

L[4, 1]

L[3, 0]

L[4, 4]

L[4, 3]

L[4, 2]

L[4, 1]

L[4, 0]

a[x, y, 63]a[x, y, 62]a[x, y, z]

Figure 11.16 SHA-3 State Matrix

5To repeat a definition from Chapter 5: If f and g are two functions, then the function F with the equation y = F(x) = g[f(x)] is called the composition of f and g and is denoted as F = g o f.

11.6 / sha-3 345

theta (θ) step

s

s

rho (ρ) step

pi (π) step

chi (χ) step

Rou

nd 0

iota (ι) step RC[0]

rot(x, y)

theta (θ) step

rho (ρ) step

pi (π) step

chi (χ) step

Rou

nd 2

3

iota (ι) step RC[23]

rot(x, y)

Figure 11.17 SHA-3 Iteration Function f

Table 11.6 Step Functions in SHA-3

Function Type Description

u Substitution New value of each bit in each word depends on its current value and on one bit in each word of preceding column and one bit of each word in succeeding column.

r Permutation The bits of each word are permuted using a circular bit shift. W[0, 0] is not affected.

p Permutation Words are permuted in the 5 * 5 matrix. W[0, 0] is not affected.

X Substitution New value of each bit in each word depends on its current value and on one bit in next word in the same row and one bit in the second next word in the same row.

ι Substitution W[0, 0] is updated by XOR with a round constant.

Boolean operations (XOR, AND, NOT) and rotations. There is no need for table lookups, arithmetic operations, or data-dependent rotations. Thus, SHA-3 is easily and efficiently implemented in either hardware or software.

We examine each of the step functions in turn.


theta SteP function The Keccak reference defines the u function as follows. For bit z in column x, row y,

u: a[x, y, z] d a[x, y, z] ⊕ a4

y== 0a[(x - 1), y, z] ⊕ a

4

y== 0a[(x + 1), y, (z - 1)] (11.1)

where the summations are XOR operations. We can see more clearly what this operation accomplishes with reference to Figure 11.18a. First, define the bitwise XOR of the lanes in column x as

C[x] = L[x, 0] ⊕ L[x, 1] ⊕ L[x, 2] ⊕ L[x, 3] ⊕ L[x, 4]

Consider lane L[x, y] in column x, row y. The first summation in Equation 11.1 performs a bitwise XOR of the lanes in column (x - 1) mod 4 to form the 64-bit lane C[x - 1]. The second summation performs a bitwise XOR of the lanes in column (x + 1) mod 4, and then rotates the bits within the 64-bit lane so that the bit in position z is mapped into position z + 1 mod 64. This forms the lane ROT

(a) θ step function

Lt[2, 3]L[2, 3] ROT(C[3], 1)C[1]

L[0, 4]

x = 0 x = 1 x = 2 x = 3 x = 4

L[0, 3]

L[0, 2]

L[0, 1]

L[0, 0]

y = 1

y = 0

y = 2

y = 3

y = 4 L[1, 4]

L[1, 3]

L[1, 2]

L[1, 1]

L[1, 0]

L[2, 4]

L[2, 3]

L[2, 2]

L[2, 1]

L[2, 0]

L[3, 4]

L[3, 3]

L[3, 2]

L[4, 1]

L[3, 0]

L[4, 4]

L[4, 3]

L[4, 2]

L[4, 1]

L[4, 0]

(b) χ step function

L[2, 3]L[2, 3] L[3, 3] AND L[4, 3]

L[0, 4]

x = 0 x = 1 x = 2 x = 3 x = 4

L[0, 3]

L[0, 2]

L[0, 1]

L[0, 0]

y = 1

y = 0

y = 2

y = 3

y = 4 L[1, 4]

L[1, 3]

L[1, 2]

L[1, 1]

L[1, 0]

L[2, 4]

L[2, 3]

L[2, 2]

L[2, 1]

L[2, 0]

L[3, 4]

L[3, 3]

L[3, 2]

L[4, 1]

L[3, 0]

L[4, 4]

L[4, 3]

L[4, 2]

L[4, 1]

L[4, 0]

Figure 11.18 Theta and Chi Step Functions

11.6 / sha-3 347

(C[x + 1], 1). These two lanes and L[x, y] are combined by bitwise XOR to form the updated value of L[x, y]. This can be expressed as

L[x, y] d L[x, y] ⊕ C[x - 1] ⊕ ROT(C[x + 1], 1)

Figure 11.18.a illustrates the operation on L[3, 2]. The same operation is per-formed on all of the other lanes in the matrix.

Several observations are in order. Each bit in a lane is updated using the bit itself and one bit in the same bit position from each lane in the preceding column and one bit in the adjacent bit position from each lane in the succeeding column. Thus the updated value of each bit depends on 11 bits. This provides good mixing. Also, the theta step provides good diffusion, as that term was defined in Chapter 3. The designers of Keccak state that the theta step provides a high level of diffusion on average and that without theta, the round function would not provide diffusion of any significance.

rho SteP function The r function is defined as follows:

r: a[x, y, z] d a[x, y, z] if x = y = 0

otherwise,

r: a[x, y, z] d a c x, y, az - (t + 1)(t + 2)

2 b d (11.2)

with t satisfying 0 … t 6 24 and a0 12 3

bt

a10b = ax

yb in GF(5)2 * 2

It is not immediately obvious what this step performs, so let us look at the process in detail.

1. The lane in position (x, y) = (0, 0), that is L[0, 0], is unaffected. For all other words, a circular bit shift within the lane is performed.

2. The variable t, with 0 … t 6 24, is used to determine both the amount of the circular bit shift and which lane is assigned which shift value.

3. The 24 individual bit shifts that are performed have the respective values (t + 1)(t + 2)

2 mod 64.

4. The shift determined by the value of t is performed on the lane in position (x, y) in the 5 * 5 matrix of lanes. Specifically, for each value of t, the corre-

sponding matrix position is defined by axyb = a0 1

2 3b

t

a10b . For example, for

t = 3, we have

axyb = a0 1

2 3b

3

a10b mod 5

= a0 12 3

ba0 12 3

ba0 12 3

ba10b mod 5


= a0 12 3

ba0 12 3

ba02b mod 5

= a0 12 3

ba26b mod 5 = a0 1

2 3ba2

1b mod 5

= a17b mod 5 = a1

2b

Table 11.7 shows the calculations that are performed to determine the amount of the bit shift and the location of each bit shift value. Note that all of the rotation amounts are different.

The r function thus consists of a simple permutation (circular shift) within each lane. The intent is to provide diffusion within each lane. Without this function, diffusion between lanes would be very slow.

Pi SteP function The p function is defined as follows:

p: a[x, y] d a[x=, y=], with axyb = a0 1

2 3bax=

y=b (11.3)

Table 11.7 Rotation Values Used in SHA-3

(a) Calculation of values and positions

t g(t) g(t) mod 64 x, y t g(t) g(t) mod 64 x, y

0 1 1 1, 0 12 91 27 4, 01 3 3 0, 2 13 105 41 0, 32 6 6 2, 1 14 120 56 3, 43 10 10 1, 2 15 136 8 4, 34 15 15 2, 3 16 153 25 3, 25 21 21 3, 3 17 171 43 2, 26 28 28 3, 0 18 190 62 2, 07 36 36 0, 1 19 210 18 0, 48 45 45 1, 3 20 231 39 4, 29 55 55 3, 1 21 253 61 2, 410 66 2 1, 4 22 276 20 4, 111 78 14 4, 4 23 300 44 1, 1

Note: g(t) = (t + 1)(t + 2)/2

axyb = a0 1

2 3b

t

a10b mod 5

(b) Rotation values by word position in matrix

x = 0 x = 1 x = 2 x = 3 x = 4y = 4 18 2 61 56 14

y = 3 41 45 15 21 8

y = 2 3 10 43 25 39

y = 1 36 44 6 55 20

y = 0 0 1 62 28 27

11.6 / sha-3 349

This can be rewritten as (x, y) * (y, (2x + 3y)). Thus, the lanes within the 5 * 5 matrix are moved so that the new x position equals the old y position and the new y position is determined by (2x + 3y) mod 5. Figure 11.19 helps in visualizing this permutation. Lanes that are along the same diagonal (increasing in y value, going from left to right) prior to p are arranged on the same row in the matrix after p is executed. Note that the position of L[0, 0] is unchanged.

Thus the p step is a permutation of lanes: The lanes move position within the 5 * 5 matrix. The r step is a permutation of bits: Bits within a lane are rotated. Note that the p step matrix positions are calculated in the same way that, for the r step, the one-dimensional sequence of rotation constants is mapped to the lanes of the matrix.

chi SteP function The x function is defined as follows:

x: a[x] d a[x] ⊕ ((a[x + 1] ⊕ 1)AND a[x + 2]) (11.4)

Z[0, 4]

x = 0 x = 1 x = 2

(a) Lane position at start of step

(b) Lane position after permutation

x = 3 x = 4

Z[0, 3]

Z[0, 2]

Z[0, 1]

Z[0, 0]

y = 1

y = 0

y = 2

y = 3

y = 4 Z[1, 4]

Z[1, 3]

Z[1, 2]

Z[1, 1]

Z[1, 0]

Z[2, 4]

Z[2, 3]

Z[2, 2]

Z[2, 1]

Z[2, 0]

Z[3, 4]

Z[3, 3]

Z[3, 2]

Z[3, 1]

Z[3, 0]

Z[4, 4]

row 0row 3

row 1row 4

row 2

row 2

row 4

row 1

row 3

Z[4, 3]

Z[4, 2]

Z[4, 1]

Z[4, 0]

Z[2, 0]

x = 0 x = 1 x = 2 x = 3 x = 4

Z[4, 0]

Z[1, 0]

Z[3, 0]

Z[0, 0]

y = 1

y = 0

y = 2

y = 3

y = 4 Z[3, 1]

Z[0, 1]

Z[2, 1]

Z[4, 1]

Z[1, 1]

Z[4, 2]

Z[1, 2]

Z[3, 2]

Z[0, 2]

Z[2, 2]

Z[0, 3]

Z[2, 3]

Z[4, 3]

Z[1, 3]

Z[3, 3]

Z[1, 4]

Z[3, 4]

Z[0, 4]

Z[2, 4]

Z[4, 4]

Figure 11.19 Pi Step Function


This function operates to update each bit based on its current value and the value of the corresponding bit position in the next two lanes in the same row. The operation is more clearly seen if we consider a single bit a[x, y, z] and write out the Boolean expression:

a[x, y, z] d a[x, y, z] ⊕ (NOT(a[x + 1, y, z]))AND(a[x + 2, y, z])

Figure 11.18b illustrates the operation of the x function on the bits of the lane L[3, 2]. This is the only one of the step functions that is a nonlinear mapping. Without it, the SHA-3 round function would be linear.

iota SteP function The i function is defined as follows:

i: a d a ⊕ RC[ir] (11.5)

This function combines an array element with a round constant that differs for each round. It breaks up any symmetry induced by the other four step functions. In fact, Equation 11.5 is somewhat misleading. The round constant is applied only to the first lane of the internal state array. We express this is as follows:

L[0, 0] d L[0, 0] ⊕ RC[ir] 0 … ir … 24

Table 11.8 lists the 24 64-bit round constants. Note that the Hamming weight, or number of 1 bits, in the round constants ranges from 1 to 6. Most of the bit posi-tions are zero and thus do not change the corresponding bits in L[0, 0]. If we take the cumulative OR of all 24 round constants, we get

RC[0] OR RC[1] OR cOR RC[23] = 800000008000808B

Thus, only 7 bit positions are active and can affect the value of L[0, 0]. Of course, from round to round, the permutations and substitutions propagate the effects of the i function to all of the lanes and all of the bit positions in the matrix. It is easily seen that the disruption diffuses through u and x to all lanes of the state after a single round.

Table 11.8 Round Constants in SHA-3

Round Constant (hexadecimal)

Number of 1 bits

Round Constant (hexadecimal)

Number of 1 bits

0 0000000000000001 1 12 000000008000808B 61 0000000000008082 3 13 800000000000008B 52 800000000000808A 5 14 8000000000008089 53 8000000080008000 3 15 8000000000008003 44 000000000000808B 5 16 8000000000008002 35 0000000080000001 2 17 8000000000000080 26 8000000080008081 5 18 000000000000800A 37 8000000000008009 4 19 800000008000000A 48 000000000000008A 3 20 8000000080008081 59 0000000000000088 2 21 8000000000008080 310 0000000080008009 4 22 0000000080000001 211 000000008000000A 3 23 8000000080008008 4



[PREN99] is a good survey of cryptographic hash functions. [GILB03] examines the secu-rity of SHA-256 through SHA-512. [CRUZ11] provides background on the development of SHA-3 and an overview of the five finalists. [PREN10] provides a good background on the cryptographic developments that led to the need for a new hash algorithm. [BURR08] dis-cusses the rationale for the new hash standard and NIST’s strategy for developing it.

BURR08 Burr, W. “A New Hash Competition.” IEEE Security & Privacy, May–June, 2008.

CRUZ11 Cruz, J. “Finding the New Encryption Standard, SHA-3.” Dr. Dobb’s, October 3, 2011. http://www.drdobbs.com/security/finding-the-new-encryption- standard-sha-/231700137

GILB03 Gilbert, H., and Handschuh, H. “Security Analysis of SHA-256 and Sisters.” Proceedings, CRYPTO ’03, 2003; published by Springer-Verlag.

PREN99 Preneel, B. “The State of Cryptographic Hash Functions.” Proceedings, EUROCRYPT ’96, 1996; published by Springer-Verlag.

PREN10 Preneel, B. “The First 30 Years of Cryptographic Hash Functions and the NIST SHA-3 Competition.” CT-RSA’10 Proceedings of the 2010 International Conference on Topics in Cryptology, 2010.

Review Questions 11.1 What characteristics are needed in a secure hash function? 11.2 What is the difference between weak and strong collision resistance? 11.3 What is the role of a compression function in a hash function?

11.8 key terms, review questiOns, and prObLems

Key Terms

absorbing phasebig endianbirthday attackbirthday paradoxbitratecapacityChi step function

collision resistantcompression functioncryptographic hash functionhash codehash functionhash value

Iota step functionkeyed hash functionKeccaklanelittle endianmessage authentication code

(MAC)MD4MD5message digestone-way hash functionPi step functionpreimage resistant

Rho step functionsecond preimage resistantSHA-1SHA-224SHA-256SHA-3SHA-384SHA-512sponge constructionsqueezing phasestrong collision resistanceTheta step functionweak collision resistance

http://www.drdobbs.com/security/finding-the-new-encryption-standard-sha-/231700137

http://www.drdobbs.com/security/finding-the-new-encryption-standard-sha-/231700137


11.4 What is the difference between little-endian and big-endian format? 11.5 What basic arithmetical and logical functions are used in SHA? 11.6 Describe the set of criteria used by NIST to evaluate SHA-3 candidates. 11.7 Define the term sponge construction. 11.8 Briefly describe the internal structure of the iteration function f. 11.9 List and briefly describe the step functions that comprise the iteration function f.

Problems 11.1 The high-speed transport protocol XTP (Xpress Transfer Protocol) uses a 32-bit

checksum function defined as the concatenation of two 16-bit functions: XOR and RXOR, defined in Section 11.4 as “two simple hash functions” and illustrated in Figure 11.5.

a. Will this checksum detect all errors caused by an odd number of error bits? Explain.

b. Will this checksum detect all errors caused by an even number of error bits? If not, characterize the error patterns that will cause the checksum to fail.

c. Comment on the effectiveness of this function for use as a hash function for authentication.

11.2 a. Consider the Davies and Price hash code scheme described in Section 11.4 and assume that DES is used as the encryption algorithm:

Hi = Hi- 1 ⊕ E(Mi, Hi- 1)

Recall the complementarity property of DES (Problem 3.14): If Y = E(K, X), then Y′ = E(K′, X′). Use this property to show how a message consisting of blocks M1, M2, c, MN can be altered without altering its hash code.

b. Show that a similar attack will succeed against the scheme proposed in [MEYE88]:

Hi = Mi ⊕ E(Hi- 1, Mi) 11.3 a. Consider the following hash function. Messages are in the form of a sequence of

numbers in Zn, M = (a1, a2, cat). The hash value h is calculated as aat

i = 1aib for

some predefined value n. Does this hash function satisfy any of the requirements for a hash function listed in Table 11.1? Explain your answer.

b. Repeat part (a) for the hash function h = aat

i = 1(ai)

2b mod n.

c. Calculate the hash function of part (b) for M = (189, 632, 900, 722, 349) and n = 989.

11.4 It is possible to use a hash function to construct a block cipher with a structure similar to DES. Because a hash function is one way and a block cipher must be reversible (to decrypt), how is it possible?

11.5 Now consider the opposite problem: using an encryption algorithm to construct a one-way hash function. Consider using RSA with a known key. Then process a message consisting of a sequence of blocks as follows: Encrypt the first block, XOR the result with the second block and encrypt again, etc. Show that this scheme is not secure by solving the following problem. Given a two-block message B1, B2, and its hash

RSAH(B1, B2) = RSA(RSA(B1) ⊕ B2)

Given an arbitrary block C1, choose C2 so that RSAH(C1, C2) = RSAH(B1, B2). Thus, the hash function does not satisfy weak collision resistance.

11.6 Suppose H(m) is a collision-resistant hash function that maps a message of arbitrary bit length into an n-bit hash value. Is it true that, for all messages x, x′ with x ≠ x′, we have H(x) ≠ H(x′) Explain your answer.


11.7 In Figure 11.12, it is assumed that an array of 80 64-bit words is available to store the values of Wt, so that they can be precomputed at the beginning of the processing of a block. Now assume that space is at a premium. As an alternative, consider the use of a 16-word circular buffer that is initially loaded with W0 through W15. Design an algorithm that, for each step t, computes the required input value Wt.

11.8 For SHA-512, show the equations for the values of W16, W17, W18, and W19. 11.9 State the value of the padding field in SHA-512 if the length of the message is a. 1919 bits b. 1920 bits c. 1921 bits 11.10 State the value of the length field in SHA-512 if the length of the message is a. 1919 bits b. 1920 bits c. 1921 bits 11.11 Suppose a1a2a3a4 are the 4 bytes in a 32-bit word. Each ai can be viewed as an integer

in the range 0 to 255, represented in binary. In a big-endian architecture, this word represents the integer

a1224 + a22

16 + a328 + a4

In a little-endian architecture, this word represents the integer

a4224 + a32

16 + a228 + a1

a. Some hash functions, such as MD5, assume a little-endian architecture. It is impor-tant that the message digest be independent of the underlying architecture. There-fore, to perform the modulo 2 addition operation of MD5 or RIPEMD-160 on a big-endian architecture, an adjustment must be made. Suppose X = x1 x2 x3 x4 and Y = y1 y2 y3 y4. Show how the MD5 addition operation (X + Y) would be carried out on a big-endian machine.

b. SHA assumes a big-endian architecture. Show how the operation (X + Y) for SHA would be carried out on a little-endian machine.

11.12 This problem introduces a hash function similar in spirit to SHA that operates on let-ters instead of binary data. It is called the toy tetragraph hash (tth).6 Given a message consisting of a sequence of letters, tth produces a hash value consisting of four letters. First, tth divides the message into blocks of 16 letters, ignoring spaces, punctuation, and capitalization. If the message length is not divisible by 16, it is padded out with nulls. A four-number running total is maintained that starts out with the value (0, 0, 0, 0); this is input to the compression function for processing the first block. The com-pression function consists of two rounds.

Round 1 Get the next block of text and arrange it as a row-wise 4 * 4 block of text and co-vert it to numbers (A = 0, B = 1, etc.). For example, for the block ABCDEFGHI-JKLMNOP, we have

A B C D

E F G H

I J K L

M N O P

6I thank William K. Mason, of the magazine staff of The Cryptogram, for providing this example.

0 1 2 3

4 5 6 7

8 9 10 11

12 13 14 15

Then, add each column mod 26 and add the result to the running total, mod 26. In this example, the running total is (24, 2, 6, 10).


Round 2 Using the matrix from round 1, rotate the first row left by 1, second row left by 2, third row left by 3, and reverse the order of the fourth row.

In our example:

B C D A

G H E F

L I J K

P O N M

1 2 3 0

6 7 4 5

11 8 9 10

15 14 13 12

Now, add each column mod 26 and add the result to the running total. The new running total is (5, 7, 9, 11). This running total is now the input into the first round of the compression function for the next block of text. After the final block is pro-cessed, convert the final running total to letters. For example, if the message is ABCDEFGHIJKLMNOP, then the hash is FHJL.

a. Draw figures comparable to Figures 11.9 and 11.10 to depict the overall tth logic and the compression function logic.

b. Calculate the hash function for the 48-letter message “I leave twenty million dol-lars to my friendly cousin Bill.”

c. To demonstrate the weakness of tth, find a 48-letter block that produces the same hash as that just derived. Hint: Use lots of A’s.

11.13 For each of the possible capacity values of SHA-3 (Table 11.5), which lanes in the internal 55 state matrix start out as lanes of all zeros?

11.14 Consider the SHA-3 option with a block size of 1024 bits and assume that each of the lanes in the first message block (P0) has at least one nonzero bit. To start, all of the lanes in the internal state matrix that correspond to the capacity portion of the initial state are all zeros. Show how long it will take before all of these lanes have at least one nonzero bit. Note: Ignore the permutation. That is, keep track of the original zero lanes even after they have changed position in the matrix.

11.15 Consider the state matrix as illustrated in Figure 11.16a. Now rearrange the rows and columns of the matrix so that L[0, 0] is in the center. Specifically, arrange the columns in the left-to-right order (x = 3, x = 4, x = 0, x = 1, x = 2) and arrange the rows in the top-to-bottom order (y = 2, y = 1, y = 0, y = 4, y = 6). This should give you some insight into the permutation algorithm used for the function and for permut-ing the rotation constants in the function. Using this rearranged matrix, describe the permutation algorithm.

11.16 The function only affects L[0, 0]. Section 11.6 states that the changes to L[0, 0] diffuse through u and to all lanes of the state after a single round.

a. Show that this is so. b. How long before all of the bit positions in the matrix are affected by the changes

to L[0, 0]?

12.1 Message Authentication Requirements

12.2 Message Authentication Functions

Message EncryptionMessage Authentication Code

12.3 Requirements for Message Authentication Codes

12.4 Security of MACs

Brute-Force AttacksCryptanalysis

12.5 MACs Based on Hash Functions: HMAC

HMAC Design ObjectivesHMAC AlgorithmSecurity of HMAC

12.6 MACs Based on Block Ciphers: DAA and CMAC

Data Authentication AlgorithmCipher-Based Message Authentication Code (CMAC)

12.7 Authenticated Encryption: CCM and GCM

Counter with Cipher Block Chaining-Message Authentication CodeGalois/Counter Mode

12.8 Key Wrapping

BackgroundThe Key Wrapping AlgorithmKey Unwrapping

12.9 Pseudorandom Number Generation Using Hash Functions and MACs

PRNG Based on Hash functionPRNG Based on MAC function



Chapter

Message authentication codes

355

356 Chapter 12 / Message authentiCation Codes

“It must have been one of those ingenious secret codes.”

—The Gloria Scott, Sir Arthur Conan Doyle

One of the most fascinating and complex areas of cryptography is that of message authentication and the related area of digital signatures. It would be impossible, in anything less than book length, to exhaust all the cryptographic functions and proto-cols that have been proposed or implemented for message authentication and digi-tal signatures. Instead, the purpose of this chapter and the next is to provide a broad overview of the subject and to develop a systematic means of describing the various approaches.

This chapter begins with an introduction to the requirements for authentica-tion and digital signature and the types of attacks to be countered. Then the basic approaches are surveyed. The remainder of the chapter deals with the fundamen-tal approach to message authentication known as the message authentication code (MAC). Following an overview of this topic, the chapter looks at security consider-ations for MACs. This is followed by a discussion of specific MACs in two categories: those built from cryptographic hash functions and those built using a block cipher mode of operation. Next, we look at a relatively recent approach known as authen-ticated encryption. Finally, we look at the use of cryptographic hash functions and MACs for pseudorandom number generation.

Learning Objectives


u List and explain the possible attacks that are relevant to message authentication.

u Define the term message authentication code.

u List and explain the requirements for a message authentication code.

u Present an overview of HMAC.

u Present an overview of CMAC.

u Explain the concept of authenticated encryption.

u Present an overview of CCM.

u Present an overview of GCM.

u Discuss the concept of key wrapping and explain its use.

u Understand how a hash function or a message authentication code can be used for pseudorandom number generation.

12.2 / Message authentiCation FunCtions 357

12.1 Message authenticatiOn requireMents

In the context of communications across a network, the following attacks can be identified.

1. Disclosure: Release of message contents to any person or process not possess-ing the appropriate cryptographic key.

2. Traffic analysis: Discovery of the pattern of traffic between parties. In a con-nection-oriented application, the frequency and duration of connections could be determined. In either a connection-oriented or connectionless environment, the number and length of messages between parties could be determined.

3. Masquerade: Insertion of messages into the network from a fraudulent source. This includes the creation of messages by an opponent that are purported to come from an authorized entity. Also included are fraudulent acknowledgments of message receipt or nonreceipt by someone other than the message recipient.

4. Content modification: Changes to the contents of a message, including inser-tion, deletion, transposition, and modification.

5. Sequence modification: Any modification to a sequence of messages between parties, including insertion, deletion, and reordering.

6. Timing modification: Delay or replay of messages. In a connection-oriented application, an entire session or sequence of messages could be a replay of some previous valid session, or individual messages in the sequence could be delayed or replayed. In a connectionless application, an individual message (e.g., datagram) could be delayed or replayed.

7. Source repudiation: Denial of transmission of message by source.

8. Destination repudiation: Denial of receipt of message by destination.

Measures to deal with the first two attacks are in the realm of message confi-dentiality and are dealt with in Part One. Measures to deal with items (3) through (6) in the foregoing list are generally regarded as message authentication. Mechanisms for dealing specifically with item (7) come under the heading of digital signatures. Generally, a digital signature technique will also counter some or all of the attacks listed under items (3) through (6). Dealing with item (8) may require a combination of the use of digital signatures and a protocol designed to counter this attack.

In summary, message authentication is a procedure to verify that received messages come from the alleged source and have not been altered. Message authen-tication may also verify sequencing and timeliness. A digital signature is an authen-tication technique that also includes measures to counter repudiation by the source.

12.2 Message authenticatiOn FunctiOns

Any message authentication or digital signature mechanism has two levels of func-tionality. At the lower level, there must be some sort of function that produces an authenticator: a value to be used to authenticate a message. This lower-level


function is then used as a primitive in a higher-level authentication protocol that enables a receiver to verify the authenticity of a message.

This section is concerned with the types of functions that may be used to pro-duce an authenticator. These may be grouped into three classes.

• Hash function: A function that maps a message of any length into a fixed-length hash value, which serves as the authenticator

• Message encryption: The ciphertext of the entire message serves as its authenticator

• Message authentication code (MAC): A function of the message and a secret key that produces a fixed-length value that serves as the authenticator

Hash functions, and how they may serve for message authentication, are dis-cussed in Chapter 11. The remainder of this section briefly examines the remaining two topics. The remainder of the chapter elaborates on the topic of MACs.

Message Encryption

Message encryption by itself can provide a measure of authentication. The analysis differs for symmetric and public-key encryption schemes.

Symmetric encryption Consider the straightforward use of symmetric encryption (Figure 12.1a). A message M transmitted from source A to destination B is en-crypted using a secret key K shared by A and B. If no other party knows the key, then confidentiality is provided: No other party can recover the plaintext of the message.

In addition, B is assured that the message was generated by A. Why? The message must have come from A, because A is the only other party that possesses K and therefore the only other party with the information necessary to construct ciphertext that can be decrypted with K. Furthermore, if M is recovered, B knows that none of the bits of M have been altered, because an opponent that does not know K would not know how to alter bits in the ciphertext to produce the desired changes in the plaintext.

So we may say that symmetric encryption provides authentication as well as confidentiality. However, this flat statement needs to be qualified. Consider exactly what is happening at B. Given a decryption function D and a secret key K, the destination will accept any input X and produce output Y = D(K, X). If X is the ci-phertext of a legitimate message M produced by the corresponding encryption func-tion, then Y is some plaintext message M. Otherwise, Y will likely be a meaningless sequence of bits. There may need to be some automated means of determining at B whether Y is legitimate plaintext and therefore must have come from A.

The implications of the line of reasoning in the preceding paragraph are pro-found from the point of view of authentication. Suppose the message M can be any arbitrary bit pattern. In that case, there is no way to determine automatically, at the destination, whether an incoming message is the ciphertext of a legitimate message. This conclusion is incontrovertible: If M can be any bit pattern, then regardless of the value of X, the value Y = D(K, X) is some bit pattern and therefore must be accepted as authentic plaintext.


Thus, in general, we require that only a small subset of all possible bit patterns be considered legitimate plaintext. In that case, any spurious ciphertext is unlikely to produce legitimate plaintext. For example, suppose that only one bit pattern in 106 is legitimate plaintext. Then the probability that any randomly chosen bit pat-tern, treated as ciphertext, will produce a legitimate plaintext message is only 10-6.

For a number of applications and encryption schemes, the desired conditions prevail as a matter of course. For example, suppose that we are transmitting English-language messages using a Caesar cipher with a shift of one (K = 1). A sends the following legitimate ciphertext:

nbsftfbupbutboeepftfbupbutboemjuumfmbnctfbujwz

B decrypts to produce the following plaintext:

mareseatoatsanddoeseatoatsandlittlelambseativy

A simple frequency analysis confirms that this message has the profile of ordi-nary English. On the other hand, if an opponent generates the following random sequence of letters:

zuvrsoevgqxlzwigamdvnmhpmccxiuureosfbcebtqxsxq


M

K K

E

(a) Symmetric encryption: confidentiality and authentication

D M

PUb

(b) Public-key encryption: confidentiality

E(K, M)

M E D M

E(PUb, M)

E(PRa, M) E(PRa, M)E(PUb, E(PRa, M))

M E D M

(c) Public-key encryption: authentication and signature

(d) Public-key encryption: confidentiality, authentication, and signature

E D

PRb

PRa

M E D M

E(PRa, M)

PRa

PUa

PUaPUb PRb

Figure 12.1 Basic Uses of Message Encryption


this decrypts to

ytuqrndufpwkyvhfzlcumlgolbbwhttqdnreabdaspwrwp

which does not fit the profile of ordinary English.It may be difficult to determine automatically if incoming ciphertext decrypts

to intelligible plaintext. If the plaintext is, say, a binary object file or digitized X-rays, determination of properly formed and therefore authentic plaintext may be difficult. Thus, an opponent could achieve a certain level of disruption simply by issuing messages with random content purporting to come from a legitimate user.

One solution to this problem is to force the plaintext to have some structure that is easily recognized but that cannot be replicated without recourse to the en-cryption function. We could, for example, append an error-detecting code, also known as a frame check sequence (FCS) or checksum, to each message before en-cryption, as illustrated in Figure 12.2a. A prepares a plaintext message M and then provides this as input to a function F that produces an FCS. The FCS is appended to M and the entire block is then encrypted. At the destination, B decrypts the incom-ing block and treats the results as a message with an appended FCS. B applies the same function F to attempt to reproduce the FCS. If the calculated FCS is equal to the incoming FCS, then the message is considered authentic. It is unlikely that any random sequence of bits would exhibit the desired relationship.

Note that the order in which the FCS and encryption functions are performed is critical. The sequence illustrated in Figure 12.2a is referred to in [DIFF79] as internal error control, which the authors contrast with external error control (Figure 12.2b). With internal error control, authentication is provided because an opponent would have difficulty generating ciphertext that, when decrypted, would have valid error control bits. If instead the FCS is the outer code, an opponent can construct messages with valid error-control codes. Although the opponent cannot

(b) External error control


K K

M | |

F

(a) Internal error control

MDF

Compare

EM

F(M) F(M)

E(K, [M || F(M)])

M | |ED

K

F

Compare

K

F

E(K, M)

F(E(K, M))

E(K, M)

M

Figure 12.2 Internal and External Error Control


know what the decrypted plaintext will be, he or she can still hope to create confu-sion and disrupt operations.

An error-control code is just one example; in fact, any sort of structuring added to the transmitted message serves to strengthen the authentication capability. Such structure is provided by the use of a communications architecture consisting of layered protocols. As an example, consider the structure of messages transmit-ted using the TCP/IP protocol architecture. Figure 12.3 shows the format of a TCP segment, illustrating the TCP header. Now suppose that each pair of hosts shared a unique secret key, so that all exchanges between a pair of hosts used the same key, regardless of application. Then we could simply encrypt all of the datagram ex-cept the IP header. Again, if an opponent substituted some arbitrary bit pattern for the encrypted TCP segment, the resulting plaintext would not include a meaning-ful header. In this case, the header includes not only a checksum (which covers the header) but also other useful information, such as the sequence number. Because successive TCP segments on a given connection are numbered sequentially, encryp-tion assures that an opponent does not delay, misorder, or delete any segments.

public-Key encryption The straightforward use of public-key encryption (Figure 12.1b) provides confidentiality but not authentication. The source (A) uses the public key PUb of the destination (B) to encrypt M. Because only B has the cor-responding private key PRb, only B can decrypt the message. This scheme provides no authentication, because any opponent could also use B’s public key to encrypt a message and claim to be A.

To provide authentication, A uses its private key to encrypt the message, and B uses A’s public key to decrypt (Figure 12.1c). This provides authentication using the same type of reasoning as in the symmetric encryption case: The message must have come from A because A is the only party that possesses PRa and therefore the only party with the information necessary to construct ciphertext that can be decrypted with PUa. Again, the same reasoning as before applies: There must be some internal structure to the plaintext so that the receiver can distinguish between well-formed plaintext and random bits.

Source port Destination port

Checksum Urgent pointer

Sequence number

Acknowledgment number

Options � padding

Application data

Reserved Flags WindowDataoffset

0Bit: 4 10 16 31

20 o

ctet

s

Figure 12.3 TCP Segment


Assuming there is such structure, then the scheme of Figure 12.1c does pro-vide authentication. It also provides what is known as digital signature.1 Only A could have constructed the ciphertext because only A possesses PRa. Not even B, the recipient, could have constructed the ciphertext. Therefore, if B is in posses-sion of the ciphertext, B has the means to prove that the message must have come from A. In effect, A has “signed” the message by using its private key to encrypt. Note that this scheme does not provide confidentiality. Anyone in possession of A’s public key can decrypt the ciphertext.

To provide both confidentiality and authentication, A can encrypt M first using its private key, which provides the digital signature, and then using B’s pub-lic key, which provides confidentiality (Figure 12.1d). The disadvantage of this ap-proach is that the public-key algorithm, which is complex, must be exercised four times rather than two in each communication.

Message Authentication Code

An alternative authentication technique involves the use of a secret key to generate a small fixed-size block of data, known as a cryptographic checksum or MAC, that is appended to the message. This technique assumes that two communicating parties, say A and B, share a common secret key K. When A has a message to send to B, it calculates the MAC as a function of the message and the key:

MAC = C(K, M)

where

M = input messageC = MAC functionK = shared secret keyMAC = message authentication code

The message plus MAC are transmitted to the intended recipient. The recipient performs the same calculation on the received message, using the same secret key, to generate a new MAC. The received MAC is compared to the calculated MAC (Figure 12.4a). If we assume that only the receiver and the sender know the identity of the secret key, and if the received MAC matches the calculated MAC, then

1. The receiver is assured that the message has not been altered. If an attacker alters the message but does not alter the MAC, then the receiver’s calculation of the MAC will differ from the received MAC. Because the attacker is as-sumed not to know the secret key, the attacker cannot alter the MAC to cor-respond to the alterations in the message.

2. The receiver is assured that the message is from the alleged sender. Because no one else knows the secret key, no one else could prepare a message with a proper MAC.

1This is not the way in which digital signatures are constructed, as we shall see, but the principle is the same.


3. If the message includes a sequence number (such as is used with HDLC, X.25, and TCP), then the receiver can be assured of the proper sequence because an attacker cannot successfully alter the sequence number.

A MAC function is similar to encryption. One difference is that the MAC algorithm need not be reversible, as it must be for decryption. In general, the MAC function is a many-to-one function. The domain of the function consists of messages of some arbitrary length, whereas the range consists of all possible MACs and all possible keys. If an n-bit MAC is used, then there are 2n possible MACs, whereas there are N possible messages with N 77 2n. Furthermore, with a k-bit key, there are 2k possible keys.

For example, suppose that we are using 100-bit messages and a 10-bit MAC. Then, there are a total of 2100 different messages but only 210 different MACs. So, on average, each MAC value is generated by a total of 2100/210 = 290 different mes-sages. If a 5-bit key is used, then there are 25 = 32 different mappings from the set of messages to the set of MAC values.

It turns out that, because of the mathematical properties of the authentication function, it is less vulnerable to being broken than encryption.

The process depicted in Figure 12.4a provides authentication but not confiden-tiality, because the message as a whole is transmitted in the clear. Confidentiality can be provided by performing message encryption either after (Figure 12.4b) or before (Figure 12.4c) the MAC algorithm. In both these cases, two separate keys are


M | |

K

C

(a) Message authentication

ME

| |

(c) Message authentication and confidentiality; authentication tied to ciphertext

M

C(K, M)

E(K2, [M || C(K1, M)])

C(K1, E(K2, M))

E(K2, M)

C

CompareK

EM | |

K1

K1

K2

K2

K2K1

K1

K2

C

(b) Message authentication and confidentiality; authentication tied to plaintext

MDC

Compare

C

C

Compare

DM

C(K1, M)

Figure 12.4 Basic Uses of Message Authentication code (MAC)


needed, each of which is shared by the sender and the receiver. In the first case, the MAC is calculated with the message as input and is then concatenated to the mes-sage. The entire block is then encrypted. In the second case, the message is encrypted first. Then the MAC is calculated using the resulting ciphertext and is concatenated to the ciphertext to form the transmitted block. Typically, it is preferable to tie the authentication directly to the plaintext, so the method of Figure 12.4b is used.

Because symmetric encryption will provide authentication and because it is widely used with readily available products, why not simply use this instead of a separate message authentication code? [DAVI89] suggests three situations in which a message authentication code is used.

1. There are a number of applications in which the same message is broadcast to a number of destinations. Examples are notification to users that the network is now unavailable or an alarm signal in a military control center. It is cheaper and more reliable to have only one destination responsible for monitoring au-thenticity. Thus, the message must be broadcast in plaintext with an associated message authentication code. The responsible system has the secret key and performs authentication. If a violation occurs, the other destination systems are alerted by a general alarm.

2. Another possible scenario is an exchange in which one side has a heavy load and cannot afford the time to decrypt all incoming messages. Authentication is carried out on a selective basis, messages being chosen at random for checking.

3. Authentication of a computer program in plaintext is an attractive service. The computer program can be executed without having to decrypt it every time, which would be wasteful of processor resources. However, if a message au-thentication code were attached to the program, it could be checked whenever assurance was required of the integrity of the program.

Three other rationales may be added.

4. For some applications, it may not be of concern to keep messages secret, but it is important to authenticate messages. An example is the Simple Network Management Protocol Version 3 (SNMPv3), which separates the functions of confidentiality and authentication. For this application, it is usually important for a managed system to authenticate incoming SNMP messages, particularly if the message contains a command to change parameters at the managed sys-tem. On the other hand, it may not be necessary to conceal the SNMP traffic.

5. Separation of authentication and confidentiality functions affords architec-tural flexibility. For example, it may be desired to perform authentication at the application level but to provide confidentiality at a lower level, such as the transport layer.

6. A user may wish to prolong the period of protection beyond the time of recep-tion and yet allow processing of message contents. With message encryption, the protection is lost when the message is decrypted, so the message is protected against fraudulent modifications only in transit but not within the target system.

Finally, note that the MAC does not provide a digital signature, because both sender and receiver share the same key.

12.3 / requireMents For Message authentiCation Codes 365

12.3 requireMents FOr Message authenticatiOn cOdes

A MAC, also known as a cryptographic checksum, is generated by a function C of the form

T = MAC(K, M)

where M is a variable-length message, K is a secret key shared only by sender and re-ceiver, and MAC(K, M) is the fixed-length authenticator, sometimes called a tag. The tag is appended to the message at the source at a time when the message is assumed or known to be correct. The receiver authenticates that message by recomputing the tag.

When an entire message is encrypted for confidentiality, using either symmet-ric or asymmetric encryption, the security of the scheme generally depends on the bit length of the key. Barring some weakness in the algorithm, the opponent must resort to a brute-force attack using all possible keys. On average, such an attack will require 2(k - 1) attempts for a k-bit key. In particular, for a ciphertext-only attack, the opponent, given ciphertext C, performs Pi = D(Ki, C) for all possible key values Ki until a Pi is produced that matches the form of acceptable plaintext.

In the case of a MAC, the considerations are entirely different. In general, the MAC function is a many-to-one function, due to the many-to-one nature of the function. Using brute-force methods, how would an opponent attempt to dis-cover a key? If confidentiality is not employed, the opponent has access to plain-text messages and their associated MACs. Suppose k 7 n; that is, suppose that the key size is greater than the MAC size. Then, given a known M1 and T1, with T1 = MAC(K, M1), the cryptanalyst can perform Ti = MAC(Ki, M1) for all pos-sible key values ki. At least one key is guaranteed to produce a match of Ti = T1. Note that a total of 2k tags will be produced, but there are only 2n 6 2k different tag values. Thus, a number of keys will produce the correct tag and the opponent has no way of knowing which is the correct key. On average, a total of 2k/2n = 2(k - n) keys will produce a match. Thus, the opponent must iterate the attack.

•Round 1

Given: M1, T1 = MAC(K, M1)Compute Ti = MAC(Ki, M1) for all 2k keysNumber of matches L 2(k - n)

•Round 2

Given: M2, T2 = MAC(K, M2)Compute Ti = MAC(Ki, M2) for the 2(k - n) keys resulting from Round 1Number of matches L 2(k - 2 * n)

And so on. On average, a rounds will be needed k = a * n. For example, if an 80-bit key is used and the tag is 32 bits, then the first round will produce about 248 possible keys. The second round will narrow the possible keys to about 216 possibili-ties. The third round should produce only a single key, which must be the one used by the sender.


If the key length is less than or equal to the tag length, then it is likely that a first round will produce a single match. It is possible that more than one key will produce such a match, in which case the opponent would need to perform the same test on a new (message, tag) pair.

Thus, a brute-force attempt to discover the authentication key is no less effort and may be more effort than that required to discover a decryption key of the same length. However, other attacks that do not require the discovery of the key are possible.

Consider the following MAC algorithm. Let M = (X1 }X2 } c}Xm) be a message that is treated as a concatenation of 64-bit blocks Xi. Then define

∆(M) = X1 ⊕ X2 ⊕ c ⊕ Xm

MAC(K, M) = E(K, ∆(M))

where ⊕ is the exclusive-OR (XOR) operation and the encryption algorithm is DES in electronic codebook mode. Thus, the key length is 56 bits, and the tag length is 64 bits. If an opponent observes {M }MAC(K, M)}, a brute-force attempt to determine K will require at least 256 encryptions. But the opponent can attack the system by replacing X1 through Xm - 1 with any desired values Y1 through Ym - 1 and replacing Xm with Ym, where Ym is calculated as

Ym = Y1 ⊕ Y2 ⊕ c ⊕ Ym - 1 ⊕ ∆(M)

The opponent can now concatenate the new message, which consists of Y1 through Ym, using the original tag to form a message that will be accepted as authen-tic by the receiver. With this tactic, any message of length 64 * (m - 1) bits can be fraudulently inserted.

Thus, in assessing the security of a MAC function, we need to consider the types of attacks that may be mounted against it. With that in mind, let us state the requirements for the function. Assume that an opponent knows the MAC func-tion but does not know K. Then the MAC function should satisfy the following requirements.

1. If an opponent observes M and MAC(K, M), it should be computationally infeasible for the opponent to construct a message M′ such that

MAC(K, M′) = MAC(K, M)

2. MAC(K, M) should be uniformly distributed in the sense that for randomly chosen messages, M and M′, the probability that MAC(K, M) = MAC(K, M′) is 2-n, where n is the number of bits in the tag.

3. Let M′ be equal to some known transformation on M. That is, M′ = f(M). For example, f may involve inverting one or more specific bits. In that case,

Pr [MAC(K, M) = MAC(K, M′)] = 2-n

The first requirement speaks to the earlier example, in which an opponent is able to construct a new message to match a given tag, even though the opponent does not know and does not learn the key. The second requirement deals with the need to thwart a brute-force attack based on chosen plaintext. That is, if we assume that the opponent does not know K but does have access to the MAC function and can present messages for MAC generation, then the opponent could try various

12.4 / seCurity oF MaCs 367

messages until finding one that matches a given tag. If the MAC function exhibits uniform distribution, then a brute-force method would require, on average, 2(n - 1) attempts before finding a message that fits a given tag.

The final requirement dictates that the authentication algorithm should not be weaker with respect to certain parts or bits of the message than others. If this were not the case, then an opponent who had M and MAC(K, M) could attempt varia-tions on M at the known “weak spots” with a likelihood of early success at produc-ing a new message that matched the old tags.

12.4 security OF Macs

Just as with encryption algorithms and hash functions, we can group attacks on MACs into two categories: brute-force attacks and cryptanalysis.

Brute-Force Attacks

A brute-force attack on a MAC is a more difficult undertaking than a brute-force attack on a hash function because it requires known message-tag pairs. Let us see why this is so. To attack a hash code, we can proceed in the following way. Given a fixed message x with n-bit hash code h = H(x), a brute-force method of finding a collision is to pick a random bit string y and check if H(y) = H(x). The attacker can do this repeatedly off line. Whether an off-line attack can be used on a MAC algorithm depends on the relative size of the key and the tag.

To proceed, we need to state the desired security property of a MAC algo-rithm, which can be expressed as follows.

• Computation resistance: Given one or more text-MAC pairs [xi, MAC(K, xi)], it is computationally infeasible to compute any text-MAC pair [x, MAC(K, x)] for any new input x ≠ xi.

In other words, the attacker would like to come up with the valid MAC code for a given message x. There are two lines of attack possible: attack the key space and attack the MAC value. We examine each of these in turn.

If an attacker can determine the MAC key, then it is possible to generate a valid MAC value for any input x. Suppose the key size is k bits and that the attacker has one known text-tag pair. Then the attacker can compute the n-bit tag on the known text for all possible keys. At least one key is guaranteed to produce the cor-rect tag, namely, the valid key that was initially used to produce the known text-tag pair. This phase of the attack takes a level of effort proportional to 2k (that is, one operation for each of the 2k possible key values). However, as was described earlier, because the MAC is a many-to-one mapping, there may be other keys that produce the correct value. Thus, if more than one key is found to produce the correct value, additional text-tag pairs must be tested. It can be shown that the level of effort drops off rapidly with each additional text-MAC pair and that the overall level of effort is roughly 2k [MENE97].

An attacker can also work on the tag without attempting to recover the key. Here, the objective is to generate a valid tag for a given message or to find a message


that matches a given tag. In either case, the level of effort is comparable to that for attacking the one-way or weak collision-resistant property of a hash code, or 2n. In the case of the MAC, the attack cannot be conducted off line without further input; the attacker will require chosen text-tag pairs or knowledge of the key.

To summarize, the level of effort for brute-force attack on a MAC algorithm can be expressed as min(2k, 2n). The assessment of strength is similar to that for symmetric encryption algorithms. It would appear reasonable to require that the key length and tag length satisfy a relationship such as min(k, n) Ú N, where N is perhaps in the range of 128 bits.

Cryptanalysis

As with encryption algorithms and hash functions, cryptanalytic attacks on MAC algorithms seek to exploit some property of the algorithm to perform some attack other than an exhaustive search. The way to measure the resistance of a MAC algo-rithm to cryptanalysis is to compare its strength to the effort required for a brute-force attack. That is, an ideal MAC algorithm will require a cryptanalytic effort greater than or equal to the brute-force effort.

There is much more variety in the structure of MACs than in hash functions, so it is difficult to generalize about the cryptanalysis of MACs. Furthermore, far less work has been done on developing such attacks. A useful survey of some methods for specific MACs is [PREN96].

12.5 Macs based On hash FunctiOns: hMac

Later in this chapter, we look at examples of a MAC based on the use of a sym-metric block cipher. This has traditionally been the most common approach to constructing a MAC. In recent years, there has been increased interest in develop-ing a MAC derived from a cryptographic hash function. The motivations for this interest are

1. Cryptographic hash functions such as MD5 and SHA generally execute faster in software than symmetric block ciphers such as DES.

2. Library code for cryptographic hash functions is widely available.

With the development of AES and the more widespread availability of code for encryption algorithms, these considerations are less significant, but hash-based MACs continue to be widely used.

A hash function such as SHA was not designed for use as a MAC and can-not be used directly for that purpose, because it does not rely on a secret key. There have been a number of proposals for the incorporation of a secret key into an existing hash algorithm. The approach that has received the most support is HMAC [BELL96a, BELL96b]. HMAC has been issued as RFC 2104, has been chosen as the mandatory-to-implement MAC for IP security, and is used in other Internet protocols, such as SSL. HMAC has also been issued as a NIST standard (FIPS 198).

12.5 / MaCs Based on hash FunCtions: hMaC 369

HMAC Design Objectives

RFC 2104 lists the following design objectives for HMAC.

• To use, without modifications, available hash functions. In particular, to use hash functions that perform well in software and for which code is freely and widely available.

• To allow for easy replaceability of the embedded hash function in case faster or more secure hash functions are found or required.

• To preserve the original performance of the hash function without incurring a significant degradation.

• To use and handle keys in a simple way.

• To have a well understood cryptographic analysis of the strength of the au-thentication mechanism based on reasonable assumptions about the embed-ded hash function.

The first two objectives are important to the acceptability of HMAC. HMAC treats the hash function as a “black box.” This has two benefits. First, an existing implementation of a hash function can be used as a module in implementing HMAC. In this way, the bulk of the HMAC code is prepackaged and ready to use without modification. Second, if it is ever desired to replace a given hash function in an HMAC implementation, all that is required is to remove the existing hash function module and drop in the new module. This could be done if a faster hash function were desired. More important, if the security of the embedded hash function were compro-mised, the security of HMAC could be retained simply by replacing the embedded hash function with a more secure one (e.g., replacing SHA@2 with SHA@3).

The last design objective in the preceding list is, in fact, the main advantage of HMAC over other proposed hash-based schemes. HMAC can be proven secure provided that the embedded hash function has some reasonable cryptographic strengths. We return to this point later in this section, but first we examine the struc-ture of HMAC.

HMAC Algorithm

Figure 12.5 illustrates the overall operation of HMAC. Define the following terms.

H = embedded hash function (e.g., MD5, SHA-1, RIPEMD-160)IV = initial value input to hash functionM = message input to HMAC (including the padding specified in the embedded

hash function)Yi = i th block of M, 0 … i … (L - 1)L = number of blocks in Mb = number of bits in a blockn = length of hash code produced by embedded hash functionK = secret key; recommended length is Ú n; if key length is greater than b, the

key is input to the hash function to produce an n-bit keyK+ = K padded with zeros on the left so that the result is b bits in length


ipad = 00110110 (36 in hexadecimal) repeated b/8 timesopad = 01011100 (5C in hexadecimal) repeated b/8 times

Then HMAC can be expressed as

HMAC(K, M) = H[(K + ⊕ opad) } H[(K + ⊕ ipad) } M]]

We can describe the algorithm as follows.

1. Append zeros to the left end of K to create a b-bit string K + (e.g., if K is of length 160 bits and b = 512, then K will be appended with 44 zeroes).

2. XOR (bitwise exclusive-OR) K + with ipad to produce the b-bit block Si.

3. Append M to Si.

4. Apply H to the stream generated in step 3.

5. XOR K + with opad to produce the b-bit block So.

6. Append the hash result from step 4 to So.

7. Apply H to the stream generated in step 6 and output the result.

Note that the XOR with ipad results in flipping one-half of the bits of K. Similarly, the XOR with opad results in flipping one-half of the bits of K, using

K�

Si

So

Y0 Y1 YL�1

b bits

b bits

b bits b bits

�

ipad

K�opad

HashIVn bits

n bits

Pad to b bits

HashIVn bits

n bits

HMAC(K, M)

H(Si || M)

�

Figure 12.5 HMAC Structure

12.5 / MaCs Based on hash FunCtions: hMaC 371

a different set of bits. In effect, by passing Si and So through the compression function of the hash algorithm, we have pseudorandomly generated two keys from K.

HMAC should execute in approximately the same time as the embedded hash function for long messages. HMAC adds three executions of the hash compression function (for Si, So, and the block produced from the inner hash).

A more efficient implementation is possible, as shown in Figure 12.6. Two quantities are precomputed:

f(IV, (K + ⊕ ipad))

f(IV, (K + ⊕ opad))

where f(cv, block) is the compression function for the hash function, which takes as arguments a chaining variable of n bits and a block of b bits and produces a chain-ing variable of n bits. These quantities only need to be computed initially and every time the key changes. In effect, the precomputed quantities substitute for the ini-tial value (IV) in the hash function. With this implementation, only one additional instance of the compression function is added to the processing normally produced

b bits b bits b bits

Precomputed Computed per message

HashIVn bits

b bits

n bits

Pad to b bits

n bits

n bits

HMAC(K, M)

f

IV

b bits

f f

K�

Si

So

Y0 Y1

ipad

K�opad

YL�1

H(Si || M)

�

�

Figure 12.6 Efficient Implementation of HMAC


by the hash function. This more efficient implementation is especially worthwhile if most of the messages for which a MAC is computed are short.

Security of HMAC

The security of any MAC function based on an embedded hash function depends in some way on the cryptographic strength of the underlying hash function. The appeal of HMAC is that its designers have been able to prove an exact rela-tionship between the strength of the embedded hash function and the strength of HMAC.

The security of a MAC function is generally expressed in terms of the prob-ability of successful forgery with a given amount of time spent by the forger and a given number of message-tag pairs created with the same key. In essence, it is proved in [BELL96a] that for a given level of effort (time, message–tag pairs) on messages generated by a legitimate user and seen by the attacker, the probability of successful attack on HMAC is equivalent to one of the following attacks on the embedded hash function.

1. The attacker is able to compute an output of the compression function even with an IV that is random, secret, and unknown to the attacker.

2. The attacker finds collisions in the hash function even when the IV is random and secret.

In the first attack, we can view the compression function as equivalent to the hash function applied to a message consisting of a single b-bit block. For this attack, the IV of the hash function is replaced by a secret, random value of n bits. An attack on this hash function requires either a brute-force attack on the key, which is a level of effort on the order of 2n, or a birthday attack, which is a special case of the second attack, discussed next.

In the second attack, the attacker is looking for two messages M and M′ that produce the same hash: H(M) = H(M′). This is the birthday attack discussed in Chapter 11. We have shown that this requires a level of effort of 2n/2 for a hash length of n. On this basis, the security of MD5 is called into question, because a level of effort of 264 looks feasible with today’s technology. Does this mean that a 128-bit hash function such as MD5 is unsuitable for HMAC? The answer is no, because of the following argument. To attack MD5, the attacker can choose any set of messages and work on these off line on a dedicated computing facility to find a collision. Because the attacker knows the hash algorithm and the default IV, the attacker can generate the hash code for each of the messages that the attacker gen-erates. However, when attacking HMAC, the attacker cannot generate message/code pairs off line because the attacker does not know K. Therefore, the attacker must observe a sequence of messages generated by HMAC under the same key and perform the attack on these known messages. For a hash code length of 128 bits, this requires 264 observed blocks (272 bits) generated using the same key. On a 1-Gbps link, one would need to observe a continuous stream of messages with no change in key for about 150,000 years in order to succeed. Thus, if speed is a con-cern, it is fully acceptable to use MD5 rather than SHA-1 as the embedded hash function for HMAC.

12.6 / MaCs Based on BloCk Ciphers: daa and CMaC 373

12.6 Macs based On bLOck ciphers: daa and cMac

In this section, we look at two MACs that are based on the use of a block cipher mode of operation. We begin with an older algorithm, the Data Authentication Algorithm (DAA), which is now obsolete. Then we examine CMAC, which is de-signed to overcome the deficiencies of DAA.

Data Authentication Algorithm

The Data Authentication Algorithm (DAA), based on DES, has been one of the most widely used MACs for a number of years. The algorithm is both a FIPS pub-lication (FIPS PUB 113) and an ANSI standard (X9.17). However, as we discuss subsequently, security weaknesses in this algorithm have been discovered, and it is being replaced by newer and stronger algorithms.

The algorithm can be defined as using the cipher block chaining (CBC) mode of operation of DES (Figure 6.4) with an initialization vector of zero. The data (e.g., message, record, file, or program) to be authenticated are grouped into contiguous 64-bit blocks: D1, D2, c, DN. If necessary, the final block is padded on the right with zeroes to form a full 64-bit block. Using the DES encryption algorithm E and a se-cret key K, a data authentication code (DAC) is calculated as follows (Figure 12.7).

O1 = E(K, D)O2 = E(K, [D2 ⊕ O1])O3 = E(K, [D3 ⊕ O2])###ON = E(K, [DN ⊕ ON - 1])

Time � 1

DESencrypt

K(56 bits)

Time � 2

K

� � �

K K

Time � NTime � N � 1

O1(64 bits)

O2

D1(64 bits)

D2 DN�1

ON

DN

ON�1

DAC(16 to 64 bits)

DESencrypt

DESencrypt

DESencrypt

Figure 12.7 Data Authentication Algorithm (FIPS PUB 113)


The DAC consists of either the entire block ON or the leftmost M bits of the block, with 16 … M … 64.

Cipher-Based Message Authentication Code (CMAC)

As was mentioned, DAA has been widely adopted in government and industry. [BELL00] demonstrated that this MAC is secure under a reasonable set of security criteria, with the following restriction. Only messages of one fixed length of mn bits are processed, where n is the cipher block size and m is a fixed positive integer. As a simple example, notice that given the CBC MAC of a one-block message X, say T = MAC(K, X), the adversary immediately knows the CBC MAC for the two-block message X } (X ⊕ T) since this is once again T.

Black and Rogaway [BLAC00] demonstrated that this limitation could be overcome using three keys: one key K of length k to be used at each step of the cipher block chaining and two keys of length b, where b is the cipher block length. This proposed construction was refined by Iwata and Kurosawa so that the two n-bit keys could be derived from the encryption key, rather than being provided separately [IWAT03]. This refinement, adopted by NIST, is the Cipher-based Message Authentication Code (CMAC) mode of operation for use with AES and triple DES. It is specified in NIST Special Publication 800-38B.

First, let us define the operation of CMAC when the message is an integer multiple n of the cipher block length b. For AES, b = 128, and for triple DES, b = 64. The message is divided into n blocks (M1, M2, c, Mn). The algorithm makes use of a k-bit encryption key K and a b-bit constant, K1. For AES, the key size k is 128, 192, or 256 bits; for triple DES, the key size is 112 or 168 bits. CMAC is calculated as follows (Figure 12.8).

C1 = E(K, M1)C2 = E(K, [M2 ⊕ C1])C3 = E(K, [M3 ⊕ C2])###Cn = E(K, [Mn ⊕ Cn - 1 ⊕ K1])T = MSBTlen(Cn)

where

T = message authentication code, also referred to as the tagTlen = bit length of TMSBs(X) = the s leftmost bits of the bit string X

If the message is not an integer multiple of the cipher block length, then the final block is padded to the right (least significant bits) with a 1 and as many 0s as necessary so that the final block is also of length b. The CMAC operation then pro-ceeds as before, except that a different b-bit key K2 is used instead of K1.

12.6 / MaCs Based on BloCk Ciphers: daa and CMaC 375

The two b-bit keys are derived from the k-bit encryption key as follows.

L = E(K, 0b)K1 = L # xK2 = L # x2 = (L # x) # x

where multiplication ( # ) is done in the finite field GF(2b) and x and x2 are first- and second-order polynomials that are elements of GF(2b). Thus, the binary rep-resentation of x consists of b - 2 zeros followed by 10; the binary representa-tion of x2 consists of b - 3 zeros followed by 100. The finite field is defined with respect to an irreducible polynomial that is lexicographically first among all such polynomials with the minimum possible number of nonzero terms. For the two approved block sizes, the polynomials are x64 + x4 + x3 + x + 1 and x128 + x7 + x2 + x + 1.

To generate K1 and K2, the block cipher is applied to the block that consists entirely of 0 bits. The first subkey is derived from the resulting ciphertext by a left shift of one bit and, conditionally, by XORing a constant that depends on the block size. The second subkey is derived in the same manner from the first subkey. This property of finite fields of the form GF(2b) was explained in the discussion of MixColumns in Chapter 5.

EncryptK K K

T

Encrypt Encrypt

MSB(Tlen)

M1

K1

K2

M2 Mn

(a) Message length is integer multiple of block size

EncryptK K K

T

Encrypt Encrypt

MSB(Tlen)

10...0

(b) Message length is not integer multiple of block size

b

k

MnM1 M2

Figure 12.8 Cipher-Based Message Authentication Code (CMAC)


12.7 authenticated encryptiOn: ccM and gcM

Authenticated encryption (AE) is a term used to describe encryption systems that simultaneously protect confidentiality and authenticity (integrity) of communica-tions. Many applications and protocols require both forms of security, but until re-cently the two services have been designed separately.

There are four common approaches to providing both confidentiality and encryption for a message M.

• Hashing followed by encryption (H S E): First compute the cryptographic hash function over M as h = H(M). Then encrypt the message plus hash func-tion: E(K, (M }h)).

• Authentication followed by encryption (A S E): Use two keys. First authen-ticate the plaintext by computing the MAC value as T = MAC(K1, M). Then encrypt the message plus tag: E(K2, [M }T ]). This approach is taken by the SSL/TLS protocols (Chapter 17).

• Encryption followed by authentication (E S A): Use two keys. First encrypt the message to yield the ciphertext C = E(K2, M). Then authenticate the ciphertext with T = MAC(K1, C) to yield the pair (C, T). This approach is used in the IPSec protocol (Chapter 20).

• Independently encrypt and authenticate (E + A). Use two keys. Encrypt the message to yield the ciphertext C = E(K2, M). Authenticate the plain-text with T = MAC(K1, M) to yield the pair (C, T). These operations can be performed in either order. This approach is used by the SSH protocol (Chapter 17).

Both decryption and verification are straightforward for each approach. For H S E, A S E, and E + A, decrypt first, then verify. For E S A,verify first, then decrypt. There are security vulnerabilities with all of these approaches. The H S E approach is used in the Wired Equivalent Privacy (WEP) protocol to protect WiFi networks. This approach had fundamental weaknesses and led to the replace-ment of the WEP protocol. [BLAC05] and [BELL00] point out that there are security concerns in each of the three encryption/MAC approaches listed above. Nevertheless, with proper design, any of these approaches can provide a high level of security. This is the goal of the two approaches discussed in this section, both of which have been standardized by NIST.

Counter with Cipher Block Chaining-Message Authentication Code

The CCM mode of operation was standardized by NIST specifically to sup-port the security requirements of IEEE 802.11 WiFi wireless local area networks (Chapter 18), but can be used in any networking application requiring authenti-cated encryption. CCM is a variation of the encrypt-and-MAC approach to authen-ticated encryption. It is defined in NIST SP 800-38C.

The key algorithmic ingredients of CCM are the AES encryption algorithm (Chapter 5), the CTR mode of operation (Chapter 6), and the CMAC authentication

12.7 / authentiCated enCryption: CCM and gCM 377

algorithm (Section 12.6). A single key K is used for both encryption and MAC algo-rithms. The input to the CCM encryption process consists of three elements.

1. Data that will be both authenticated and encrypted. This is the plaintext mes-sage P of data block.

2. Associated data A that will be authenticated but not encrypted. An example is a protocol header that must be transmitted in the clear for proper protocol operation but which needs to be authenticated.

3. A nonce N that is assigned to the payload and the associated data. This is a unique value that is different for every instance during the lifetime of a pro-tocol association and is intended to prevent replay attacks and certain other types of attacks.

Figure 12.9 illustrates the operation of CCM. For authentication, the input includes the nonce, the associated data, and the plaintext. This input is formatted as a sequence of blocks B0 through Br. The first block contains the nonce plus some formatting bits that indicate the lengths of the N, A, and P elements. This is fol-lowed by zero or more blocks that contain A, followed by zero of more blocks that contain P. The resulting sequence of blocks serves as input to the CMAC algorithm, which produces a MAC value with length Tlen, which is less than or equal to the block length (Figure 12.9a).

For encryption, a sequence of counters is generated that must be independent of the nonce. The authentication tag is encrypted in CTR mode using the single counter Ctr0. The Tlen most significant bits of the output are XORed with the tag to produce an encrypted tag. The remaining counters are used for the CTR mode en-cryption of the plaintext (Figure 6.7). The encrypted plaintext is concatenated with the encrypted tag to form the ciphertext output (Figure 12.9b).

SP 800-38C defines the authentication/encryption process as follows.

1. Apply the formatting function to (N, A, P) to produce the blocks B0, B1, c, Br.

2. Set Y0 = E(K, B0). 3. For i = 1 to r, do Yi = E(K, (Bi ⊕ Yi- 1)). 4. Set T = MSBTlen(Yr). 5. Apply the counter generation function to generate the counter blocks

Ctr0, Ctr1, c, Ctrm, where m = <Plen/128=. 6. For j = 0 to m, do Sj = E(K, Ctrj). 7. Set S = S1 }S2 } c}Sm. 8. Return C = (P ⊕ MSBPlen(S)) } (T ⊕ MSBTlen(S0)).

For decryption and verification, the recipient requires the following input: the ciphertext C, the nonce N, the associated data A, the key K, and the initial counter Ctr0. The steps are as follows.

1. If Clen … Tlen, then return INVALID. 2. Apply the counter generation function to generate the counter blocks

Ctr0, Ctr1, c, Ctrm, where m = <Clen/128= . 3. For j = 0 to m, do Sj = E(K, Ctrj). 4. Set S = S1 }S2 } c}Sm.


5. Set P = MSBClen - Tlen(C) ⊕ MSBClen - Tlen(S).

6. Set T = LSBTlen(C) ⊕ MSBTlen(S0).

7. Apply the formatting function to (N, A, P) to produce the blocks B0, B1, c, Br.

8. Set Y0 = E(K, B0).

9. For i = 1 to r, do Yi = E(K, (Bi ⊕ Yi- 1)).

10. If T ≠ MSBTlen (Yr), then return INVALID, else return P.

(a) Authentication

(b) Encryption

B0

Ctr0

B1 B2 Br

Tag

Tag

Nonce Plaintext

Plaintext

Ciphertext

Ass. Data

K CMAC

MSB(Tlen)K

CTRCtr1, Ctr2, ..., Ctrm

EncryptK

Figure 12.9 Counter with Cipher Block Chaining-Message Authentication Code (CCM)


CCM is a relatively complex algorithm. Note that it requires two complete passes through the plaintext, once to generate the MAC value, and once for encryp-tion. Further, the details of the specification require a tradeoff between the length of the nonce and the length of the tag, which is an unnecessary restriction. Also note that the encryption key is used twice with the CTR encryption mode: once to generate the tag and once to encrypt the plaintext plus tag. Whether these com-plexities add to the security of the algorithm is not clear. In any case, two analyses of the algorithm ([JONS02] and [ROGA03]) conclude that CCM provides a high level of security.

Galois/Counter Mode

The GCM mode of operation, standardized by NIST in NIST SP 800-38D, is de-signed to be parallelizable so that it can provide high throughput with low cost and low latency. In essence, the message is encrypted in variant of CTR mode. The re-sulting ciphertext is multiplied with key material and message length information over GF(2128) to generate the authenticator tag. The standard also specifies a mode of operation that supplies the MAC only, known as GMAC.

The GCM mode makes use of two functions: GHASH, which is a keyed hash function, and GCTR, which is essentially the CTR mode with the counters deter-mined by a simple increment by one operation.

GHASHH(X) takes a input the hash key H and a bit string X such that len(X) = 128m bits for some positive integer m and produces a 128-bit MAC value. The function may be specified as follows (Figure 12.10a).

1. Let X1, X2, c, Xm - 1, Xm denote the unique sequence of blocks such that X = X1 }X2 } c}Xm - 1 }Xm.

2. Let Y0 be a block of 128 zeros, designated as 0128.

3. For i = 1, c, m, let Yi = (Yi- 1 ⊕ Xi) # H, where # designates multiplication in GF(2128).

4. Return Ym.

The GHASHH(X) function can be expressed as

(X1# Hm) ⊕ (X2

# Hm - 1) ⊕ c ⊕ (Xm - 1# H2) ⊕ (Xm

# H)

This formulation has desirable performance implications. If the same hash key is to be used to authenticate multiple messages, then the values H2, H3, c can be precalculated one time for use with each message to be authenticated. Then, the blocks of the data to be authenticated (X1, X2, c, Xm) can be processed in parallel, because the computations are independent of one another.

GCTRK(ICB, X) takes a input a secret key K and a bit string X arbitrary length and returns a ciphertext Y of bit length len(X). The function may be speci-fied as follows (Figure 12.10b).

1. If X is the empty string, then return the empty string as Y.

2. Let n = <(len(X)/128)= . That is, n is the smallest integer greater than or equal to len(X)/128.


3. Let X1, X2, c, Xn - 1, Xn* denote the unique sequence of bit strings such that

X = X1 }X2 } c}Xn - 1 }Xn*;

X1, X2, c, Xn - 1 are complete 128-bit blocks.

4. Let CB1 = ICB.

5. For, i = 2 to n let CBi = inc32(CBi- 1), where the inc32(S) function increments the rightmost 32 bits of S by 1 mod 232, and the remaining bits are unchanged.

6. For i = 1 to n - 1, do Yi = Xi ⊕ E(K, CBi).

7. Let Y n* = Xn

* ⊕ MSBlen(Xn*)(E(K, CBn)).

8. Let Y = Y1 }Y2 }c}Yn - 1 }Y n*

9. Return Y.

Note that the counter values can be quickly generated and that the encryption operations can be performed in parallel.

(a) GHASHH(X1 || X2 || . . . || Xm) � Ym

X1

X1

X2

ICB

Xm

Y1

Y1

Y2 Ym

H

E

inc

H H

K

X2

CB2

Y2

EK

Xn–1

CBn–1

Yn–1

E

inc

K

Xn

CBn

Yn

E

MSB

K

*

(b) GCTRK(ICB, X1 || X2 || . . . || Xn) � Y1 || Y2 || . . . ||Yn**

*

Figure 12.10 GCM Authentication and Encryption Functions


We can now define the overall authenticated encryption function (Figure 12.11). The input consists of a secret key K, an initialization vector IV , a plaintext P, and additional authenticated data A. The notation [x]s means the s-bit binary representation of the nonnegative integer x. The steps are as follows.

1. Let H = E(K, 0128).

2. Define a block, J0, as

If len(IV) = 96, then let J0 = IV }031 }1.

If len (IV) ≠ 96, then let s = 128 <len(IV)/128= - len(IV), and let

J0 = GHASHH(IV }0s + 64 } [len(IV)]64).

3. Let C = GCTRK(inc32(J0), P).

4. Let u = 128 < len(C)/128 = - len(C) and let v = 128 <len(A)/128= - len(A).

5. Define a block, S, as

S = GHASHH(A }0v }C }0u } [len(A)]64 } [len(C)]64)

IV

J0

J0

Plaintext

K

K

GCTR

encode

incr

GCTR

Tag

GHASH

A = Ass. Data C = Ciphertext [len(A)]64 [len(C)]640v 0u

MSBt

EK

H

0

Figure 12.11 Galois Counter—Message Authentication Code (GCM)


6. Let T = MSBt(GCTRK(J0, S)), where t is the supported tag length.

7. Return (C, T).

In step 1, the hash key is generated by encrypting a block of all zeros with the secret key K. In step 2, the pre-counter block (J0) is generated from the IV. In particular, when the length of the IV is 96 bits, then the padding string 031 }1 is appended to the IV to form the pre-counter block. Otherwise, the IV is padded with the minimum number of 0 bits, possibly none, so that the length of the result-ing string is a multiple of 128 bits (the block size); this string in turn is appended with 64 additional 0 bits, followed by the 64-bit representation of the length of the IV, and the GHASH function is applied to the resulting string to form the pre-counter block.

Thus, GCM is based on the CTR mode of operation and adds a MAC that au-thenticates both the message and additional data that requires only authentication. The function that computes the hash uses only multiplication in a Galois field. This choice was made because the operation of multiplication is easy to perform within a Galois field and is easily implemented in hardware [MCGR05].

[MCGR04] examines the available block cipher modes of operation and shows that a CTR-based authenticated encryption approach is the most efficient mode of operation for high-speed packet networks. The paper further demonstrates that GCM meets a high level of security requirements.

12.8 key Wrapping

Background

The most recent block cipher mode of operation defined by NIST is the Key Wrap (KW) mode of operation (SP 800-38F), which uses AES or triple DEA as the un-derlying encryption algorithm. The AES version is also documented in RFC 3394.

The purpose of key wrapping is to securely exchange a symmetric key to be shared by two parties, using a symmetric key already shared by those parties. The latter key is called a key encryption key (KEK).

Two questions need to be addressed at this point. First, why do we need to use a symmetric key already known to two parties to encrypt a new symmetric key? Such a requirement is found in a number of protocols described in this book, such as the key management portion of IEEE 802.11 and IPsec. Quite often, a protocol calls for a hierarchy of keys, with keys lower on the hierarchy used more frequently, and changed more frequently to thwart attacks. A higher-level key, which is used in-frequently and therefore more resistant to cryptanalysis, is used to encrypt a newly created lower-level key so that it can be exchanged between parties that share the higher-level key.

The second question is, why do we need a new mode? The intent of the new mode is to operate on keys whose length is greater than the block size of the encryp-tion algorithm. For example, AES uses a block size of 128 bits but can use a key size of 128, 192, or 256 bits. In the latter two cases, encryption of the key involves

12.8 / key Wrapping 383

multiple blocks. We consider the value of key data to be greater than the value of other data, because the key will be used multiple times, and compromise of the key compromises all of the data encrypted with the key. Therefore, NIST desired a robust encryption mode. KW is robust in the sense that each bit of output can be expected to depend in a nontrivial fashion on each bit of input. This is not the case for any of the other modes of operation that we have described. For example, in all of the modes so far described, the last block of plaintext only influences the last block of ciphertext. Similarly, the first block of ciphertext is derived only from the first block of plaintext.

To achieve this robust operation, KW achieves a considerably lower through-put than the other modes, but the tradeoff may be appropriate for some key management applications. Also, KW is only used for small amounts of plaintext compared to, say, the encryption of a message or a file.

The Key Wrapping Algorithm

The key wrapping algorithm operates on blocks of 64 bits. The input to the algo-rithm consists of a 64-bit constant, discussed subsequently, and a plaintext key that is divided into blocks of 64 bits. We use the following notation:

MSB64(W) most significant 64 bits of W

LSB64(W) least significant 64 bits of W

W temporary value; output of encryption function

⊕ bitwise exclusive-OR

} concatenation

K key encryption key

n number of 64-bit key data blocks

s number of stages in the wrapping process; s = 6n

Pi ith plaintext key data block; 1 … i … n

Ci ith ciphertext data block; 0 … i … n

A(t) 64-bit integrity check register after encryption stage t; 1 … t … s

A(0) initial integrity check value (ICV); in hexadecimal: A6A6A6A6A6A6A6A6

R(t, i) 64-bit register i after encryption stage t; 1 … t … s; 1 … i … n

We now describe the key wrapping algorithm:

Inputs: Plaintext, n 64-bit values (P1, P2, . . . , Pn) Key encryption key, K

Outputs: Ciphertext, (n + 1) 64-bit values (C0, C1, . . . , Cn)

1. Initialize variables.

A(0) = A6A6A6A6A6A6A6A6

for i = 1 to n

R(0, i) = Pi


2. Calculate intermediate values.

for t = 1 to s

W = E(K, [A(t−1) }R(t−1, 1)])A(t) = t ⊕ MSB64(W)

R(t, n) = LSB64(W)

for i = 1 to n−1

R(t, i) = R(t−1, i+1)

3. Output results.

C0 = A(s)

for i = 1 to n

Ci = R(s, i)

Note that the ciphertext is one block longer than the plaintext key, to ac-commodate the ICV. Upon unwrapping (decryption), both the 64-bit ICV and the plaintext key are recovered. If the recovered ICV differs from the input value of hexadecimal A6A6A6A6A6A6A6A6, then an error or alteration has been detected and the plaintext key is rejected. Thus, the key wrap algorithm provides not only confidentiality but also data integrity.

Figure 12.12 illustrated the key wrapping algorithm for encrypting a 256-bit key. Each box represents one encryption stage (one value of t). Note that the A output is fed as input to the next stage (t + 1), whereas the R output skips forward n stages (t + n), which in this example is n = 4. This arrangement further increases the avalanche effect and the mixing of bits. To achieve this skipping of stages, a sliding buffer is used, so that the R output from stage t is shifted in the buffer one position for each stage, until it becomes the input for stage t + n. This might be clearer if we expand the inner for loop for a 256-bit key (n = 4). Then the assignments are as follows:

R(t, 1) = R(t - 1, 2)

R(t, 2) = R(t - 1, 3)

R(t, 3) = R(t - 1, 4)

For example, consider that at stage 5, the R output has a value of R(5, 4) = x. At stage 6, we execute R(6, 3) = R(5, 4) = x. At stage 7, we execute R(7, 2) = R (6, 3) = x. At stage 8, we execute R(8, 1) = R(7, 2) = x. So, at stage 9, the input value of R(t - 1, 1) is R(8, 1) = x.

Figure 12.13 depicts the operation of stage t for a 256-bit key. The dashed feedback lines indicate the assignment of new values to the stage variables.

Key Unwrapping

The key unwrapping algorithm can be defined as follows:

Inputs: Ciphertext, (n + 1) 64-bit values (C0, C1, . . . , Cn) Key encryption key, K

Outputs: Plaintext, n 64-bit values (P1, P2, . . . , Pn), ICV

12.8 / key Wrapping 385

t = 1 t = 2 t = 3 t = 4

t = 5 t = 6 t = 7 t = 8

t = 9 t = 10 t = 11 t = 12

t = 13 t = 14 t = 15 t = 16

t = 17 t = 18 t = 19 t = 20

t = 21

C0 = A(24)

A(0) A(1)

C1 = R(24, 1)= R(21, 4)

P1 =R(0, 1)

P2 =R(0, 2)

P3 =R(0, 3)

P4 =R(0, 4) A(2) A(3)

C2 = R(24, 2)= R(22, 4)

C3 = R(24, 3)= R(23, 4)

C4 = R(24, 4)

t = 22 t = 23 t = 24

Figure 12.12 Key Wrapping Operation for 256-Bit Key


A(s) = C0for i = 1 to n

R(s, i) = Ci

2. Calculate intermediate values.

for t = s to 1

W = D(K, [(A(t)⊕ t)} R(t, n)])


A(t−1) = MSB64(W)

R(t−1, 1) = LSB64(W)

for i = 2 to n

R(t−1, i) = R(t, i−1)

3. Output results.

if A(0) = A6A6A6A6A6A6A6A6then for i = 1 to n P(i) = R(0, i)else return error

Note that the decryption function is used in the unwrapping algorithm.We now demonstrate that the unwrap function is the inverse of the wrap func-

tion, that is, that the unwrap function recovers the plaintext key and the ICV. First, note that because the index variable t is counted down from s to 1 for unwrapping, stage t of the unwrap algorithm corresponds to stage t of the wrap algorithm. The input variables to stage t of the wrap algorithm are indexed at t - 1 and the output variables of stage t of the unwrap algorithm are indexed at t - 1. Thus, to demon-strate that the two algorithms are inverses of each other, we need only demonstrate that the output variables of stage t of the unwrap algorithm are equal to the input variables to stage t of the wrap algorithm.

This demonstration is in two parts. First we demonstrate that the calculation of A and R variables prior to the for loop are inverses. To do this, let us simplify the notation a bit. Define the 128-bit value T to be the 64-bit value t followed by 64 zeros. Then, the first three lines of step 2 of the wrap algorithm can be written as the following single line:

A(t) }R(t, n) = T ⊕ E(K, [A(t - 1) }R(t - 1, 1)]) (12.1)

The first three lines of step 2 of the unwrap algorithm can be written as:

A(t - 1) }R(t - 1, 1) = D(K, ([A(t) }R(t, n)] ⊕ T)) (12.2)

A(t – 1)

Encrypt

MSB

K

t LSB

R(t – 1, 1) R(t – 1, 2)

R(t – 1, 3)

R(t – 1, 4)

Figure 12.13 Key Wrapping Operation for 256-Bit Key: Stage t

12.9 / pseudorandoM nuMBer generation using hash FunCtions 387

Expanding the right-hand side by substituting from Equation 12.1,

D(K, ([A(t) }R(t, n)] ⊕ T)) = D(K, ([T ⊕ E(K, [A(t - 1) }R(t - 1, 1)])] ⊕ T))

Now we recognize that T ⊕ T = 0 and that for any x, x ⊕ 0 = x. So,

D(K, ([A(t) }R(t, n)] ⊕ T)) = D(K, ([E(K, [A(t - 1) }R(t - 1, 1)]))

= A(t - 1) }R(t - 1, 1)

The second part of the demonstration is to show that the for loops in step 2 of the wrap and unwrap algorithms are inverses. For stage k of the wrap algo-rithm, the variables R(t - 1, 1) through R(t - 1, n) are input. R(t - 1, 1) is used in the encryption calculation. R(t - 1, 2) through R(t - 1, n) are mapped, respectively into R(t, 1) through R(t, n - 1), and R(t, n) is output from the encryption function. For stage k of the unwrap algorithm, the variables R(t, 1) through R(t, n) are input. R(t, n) is input to the decryption function to produce R(t - 1, 1). The remaining vari-ables R(t - 1, 2) through R(t - 1, n) are generated by the for loop, such that they are mapped, respectively, from R(t, 1) through R(t, n - 1).

Thus, we have shown that the output variables of stage k of the unwrap algo-rithm equal the input variables of stage k of the wrap algorithm.

12.9 pseudOrandOM nuMber generatiOn using hash FunctiOns and Macs

The essential elements of any pseudorandom number generator (PRNG) are a seed value and a deterministic algorithm for generating a stream of pseudorandom bits. If the algorithm is used as a pseudorandom function (PRF) to produce a required value, such as a session key, then the seed should only be known to the user of the PRF. If the algorithm is used to produce a stream encryption function, then the seed has the role of a secret key that must be known to the sender and the receiver.

We noted in Chapters 7 and 10 that, because an encryption algorithm pro-duces an apparently random output, it can serve as the basis of a (PRNG). Similarly, a hash function or MAC produces apparently random output and can be used to build a PRNG. Both ISO standard 18031 (Random Bit Generation) and NIST SP 800-90 (Recommendation for Random Number Generation Using Deterministic Random Bit Generators) define an approach for random number generation using a cryptographic hash function. SP 800-90 also defines a random number generator based on HMAC. We look at these two approaches in turn.

PRNG Based on Hash Function

Figure 12.14a shows the basic strategy for a hash-based PRNG specified in SP 800-90 and ISO 18031. The algorithm takes as input:

V = seed

seedlen = bit length of V Ú k + 64, where k is a desired security level expressed in bits

n = desired number of output bits


The algorithm uses the cryptographic hash function H with an hash value out-put of outlen bits. The basic operation of the algorithm is

m = <n/outlen=data = VW = the null stringFor i = 1 to m

wi = H(data)W = W || wi

data = (data + 1) mod 2seedlen

Return leftmost n bits of W

Thus, the pseudorandom bit stream is w1 }w2 }c}wm with the final block truncated if required.

The SP 800-90 specification also provides for periodically updating V to en-hance security. The specification also indicates that there are no known or suspected weaknesses in the hash-based approach for a strong cryptographic hash algorithm, such as SHA-2.

(a) PRNG using cryptographic hash function

(b) PRNG using HMAC

V

K

Cryptographic hash function

Pseudorandomoutput

+1

V

HMAC

Pseudorandomoutput

Figure 12.14 Basic Structure of Hash-Based PRNGs (SP 800-90)

12.9 / pseudorandoM nuMBer generation using hash FunCtions 389

PRNG Based on MAC Function

Although there are no known or suspected weaknesses in the use of a cryptographic hash function for a PRNG in the manner of Figure 12.14a, a higher degree of con-fidence can be achieved by using a MAC. Almost invariably, HMAC is used for constructing a MAC-based PRNG. This is because HMAC is a widely used stan-dardized MAC function and is implemented in many protocols and applications. As SP 800-90 points out, the disadvantage of this approach compared to the hash-based approach is that the execution time is twice as long, because HMAC involves two executions of the underlying hash function for each output block. The advan-tage of the HMAC approach is that it provides a greater degree of confidence in its security, compared to a pure hash-based approach.

For the MAC-based approach, there are two inputs: a key K and a seed V. In effect, the combination of K and V form the overall seed for the PRNG specified in SP 800-90. Figure 12.14b shows the basic structure of the PRNG mechanism, and the leftmost column of Figure 12.15 shows the logic. Note that the key remains the same for each block of output, and the data input for each block is equal to the tag output of the previous block. The SP 800-90 specification also provides for periodi-cally updating K and V to enhance security.

It is instructive to compare the SP 800-90 recommendation with the use of HMAC for a PRNG in some applications, and this is shown in Figure 12.15. For the IEEE 802.11i wireless LAN security standard (Chapter 18), the data input consists of the seed concatenated with a counter. The counter is incremented for each block wi of output. This approach would seem to offer enhanced security compared to the SP 800-90 approach. Consider that for SP 800-90, the data input for output block wi is just the output wi- 1 of the previous execution of HMAC. Thus, an opponent who is able to observe the pseudorandom output knows both the input and output of HMAC. Even so, with the assumption that HMAC is secure, knowledge of the input and output should not be sufficient to recover K and hence not sufficient to predict future pseudorandom bits.

The approach taken by the Transport Layer Security protocol (Chapter 17) and the Wireless Transport Layer Security Protocol (Chapter 18) involves invoking HMAC twice for each block of output wi. As with IEEE 802.11, this is done in such a way that the output does not yield direct information about the input. The double use of HMAC doubles the execution burden and would seem to be security overkill.

m = <n/outlen=w0 = VW = the null stringFor i = 1 to m

wi = MAC(K, wi- 1)W = W }wi


m = <n/outlen=W = the null stringFor i = 1 to m

wi = MAC(K, (V } i))W = W }wi


m = <n/outlen=A(0) = VW = the null stringFor i = 1 to mA(i) = MAC(K, A(i - 1))wi = MAC(K, (A(i) }V)W = W }wi


NIST SP 800-90 IEEE 802.11i TLS/WTLS

Figure 12.15 Three PRNGs Based on HMAC


12.11 key terMs, revieW questiOns, and prObLeMs

Key Terms

authenticatorCipher-Based Message

Authentication Code (CMAC)

CMACCounter with Cipher Block

Chaining-Message Authentication Code (CCM)

cryptographic checksumcryptographic hash

functionData Authentication

Algorithm (DAA)Galois/Counter Mode

(GCM)HMAC

key encryption keyKey Wrap modekey wrappingmessage authenticationmessage authentication code

(MAC)

BELL96a Bellare, M.; Canetti, R.; and Krawczyk, H. “Keying Hash Functions for Message Authentication.” Proceedings, CRYPTO ’96, August 1996; published by Springer-Verlag. An expanded version is available at http://www-cse.ucsd.edu/users/mihir

BELL96b Bellare, M.; Canetti, R.; and Krawczyk, H. “The HMAC Construction.” CryptoBytes, Spring 1996.

JUEN85 Jueneman, R.; Matyas, S.; and Meyer, C. “Message Authentication.” IEEE Communications Magazine, September 1988.

JUEN87 Jueneman, R. “Electronic Document Authentication.” IEEE Network Magazine, April 1987.

Review Questions 12.1 What types of attacks are addressed by message authentication? 12.2 What two levels of functionality comprise a message authentication or digital signa-

ture mechanism? 12.3 What are some approaches to producing message authentication? 12.4 When a combination of symmetric encryption and an error control code is used for

message authentication, in what order must the two functions be performed? 12.5 What is a message authentication code? 12.6 What is the difference between a message authentication code and a one-way hash

function?


[JUEN85] and [JUEN87] provide a good background on message authentication with a focus on cryptographic MACs and hash functions. Overviews of HMAC can be found in [BELL96a] and [BELL96b].

http://www-cse.ucsd.edu/users/mihir


12.11 / key terMs, revieW questions, and proBleMs 391

12.7 In what ways can a hash value be secured so as to provide message authentication? 12.8 Is it necessary to recover the secret key in order to attack a MAC algorithm? 12.9 What changes in HMAC are required in order to replace one underlying hash func-

tion with another?

Problems 12.1 If F is an error-detection function, either internal or external use (Figure 12.2) will

provide error-detection capability. If any bit of the transmitted message is altered, this will be reflected in a mismatch of the received FCS and the calculated FCS, whether the FCS function is performed inside or outside the encryption function. Some codes also provide an error-correction capability. Depending on the nature of the function, if one or a small number of bits is altered in transit, the error-correction code contains sufficient redundant information to determine the errored bit or bits and correct them. Clearly, an error-correction code will provide error correction ca-pability when used external to the encryption function. Will it also provide this capa-bility if used internal to the encryption function?

12.2 The data authentication algorithm, described in Section 12.6, can be defined as using the cipher block chaining (CBC) mode of operation of DES with an initialization vec-tor of zero (Figure 12.7). Show that the same result can be produced using the cipher feedback mode.

12.3 At the beginning of Section 12.6, it was noted that given the CBC MAC of a one-block message X , say T = MAC(K, X), the adversary immediately knows the CBC MAC for the two-block message X || (X ⊕ T) since this is once again T . Justify this statement.

12.4 In this problem, we demonstrate that for CMAC, a variant that XORs the second key after applying the final encryption doesn’t work. Let us consider this for the case of the message being an integer multiple of the block size. Then, the variant can be expressed as VMAC(K, M) = CBC(K, M) ⊕ K1. Now suppose an adver-sary is able to ask for the MACs of three messages: the message 0 = 0n, where n is the cipher block size; the message 1 = 1n; and the message 1 } 0. As a result of these three queries, the adversary gets T0 = CBC(K, 0) ⊕ K1; T1 = CBC(K, 1) ⊕ K1 and T2 = CBC(K, [CBC(K, 1)]) ⊕ K1. Show that the adversary can compute the correct MAC for the (unqueried) message 0 } (T0 ⊕ T1).

12.5 In the discussion of subkey generation in CMAC, it states that the block cipher is ap-plied to the block that consists entirely of 0 bits. The first subkey is derived from the resulting string by a left shift of one bit and, conditionally, by XORing a constant that depends on the block size. The second subkey is derived in the same manner from the first subkey.

a. What constants are needed for block sizes of 64 and 128 bits? b. Explain how the left shift and XOR accomplishes the desired result. 12.6 Section 12.6 listed three general approaches to authenticated encryption: A S E,

E S A, E + A. a. Which approach is used by CCM? b. Which approach is used by GCM? 12.7 Show that the GHASH function calculates

(X1# Hm) ⊕ (X2

# Hm - 1) ⊕ c⊕ (Xm - 1# H2) ⊕ (Xm

# H)

12.8 Draw a figure similar to Figure 12.11 that shows authenticated decryption. 12.9 Alice want to send a single bit of information (a yes or a no) to Bob by means of a

word of length 2. Alice and Bob have four possible keys available to perform message


authentication. The following matrix shows the 2-bit word sent for each message under each key:

Message

Key 0 1

1 00 01

2 10 00

3 01 11

4 11 10

a. The preceding matrix is in a useful form for Alice. Construct a matrix with the same information that would be more useful for Bob.

b. What is the probability that someone else can successfully impersonate Alice? c. What is the probability that someone can replace an intercepted message with

another message successfully? 12.10 Draw figures similar to Figures 12.12 and 12.13 for the unwrap algorithm. 12.11 Consider the following key wrapping algorithm:

1. Initialize variables.A = A6A6A6A6A6A6A6A6for i = 1 to n R(i) = Pi

2. Calculate intermediate values.for j = 0 to 5 for i = 1 to n B = E(K, [A || R(i)]) t = (n × j)+i A = t ⊕ MSB64(B) R(i) = LSB64(B)

3. Output results.C0 = Afor i = 1 to n Ci = R(i)

a. Compare this algorithm, functionally, with the algorithm specified in SP 800-38F and described in Section 12.8.

b. Write the corresponding unwrap algorithm.

13.1 Digital Signatures

PropertiesAttacks and ForgeriesDigital Signature RequirementsDirect Digital Signature

13.2 Elgamal Digital Signature Scheme

13.3 Schnorr Digital Signature Scheme

13.4 NIST Digital Signature Algorithm

The DSA ApproachThe Digital Signature Algorithm

13.5 Elliptic Curve Digital Signature Algorithm

Global Domain ParametersKey Generation Digital Signature Generation and Authentication

13.6 RSA-PSS Digital Signature Algorithm

Mask Generation FunctionThe Signing OperationSignature Verification



Chapter

Digital SignatureS

393

394 Chapter 13 / Digital SignatureS

To guard against the baneful influence exerted by strangers is therefore an elementary dictate of savage prudence. Hence before strangers are allowed to enter a district, or at least before they are permitted to mingle freely with the inhabitants, certain ceremonies are often performed by the natives of the country for the purpose of disarming the strangers of their magical powers, or of disinfecting, so to speak, the tainted atmosphere by which they are supposed to be surrounded.


Learning Objectives


u Present an overview of the digital signature process.

u Understand the Elgamal digital signature scheme.

u Understand the Schnorr digital signature scheme.

u Understand the NIST digital signature scheme.

u Compare and contrast the NIST digital signature scheme with the Elgamal and Schnorr digital signature schemes.

u Understand the elliptic curve digital signature scheme.

u Understand the RSA-PSS digital signature scheme.

The most important development from the work on public-key cryptography is the digital signature. The digital signature provides a set of security capabilities that would be difficult to implement in any other way.

Figure 13.1 is a generic model of the process of making and using digital sig-natures. Bob can sign a message using a digital signature generation algorithm. The inputs to the algorithm are the message and Bob’s private key. Any other user, say Alice, can verify the signature using a verification algorithm, whose inputs are the message, the signature, and Bob’s public key. In simplified terms, the essence of the digital signature mechanism is shown in Figure 13.2. This repeats the logic shown in Figure 11.4. A worked-out example, using RSA, is available at this book’s Premium Content Web site.

We begin this chapter with an overview of digital signatures. We then present the Elgamal and Schnorr digital signature schemes, understanding of which makes it easier to understand the Digital Signature Algorithm (DSA). The chapter then cov-ers the two other important standardized digital signature schemes: the Elliptic Curve Digital Signature Algorithm (ECDSA) and the RSA Probabilistic Signature Scheme (RSA-PSS).

13.1 / Digital SignatureS 395

13.1 DigitaL signatures

Properties

Message authentication protects two parties who exchange messages from any third party. However, it does not protect the two parties against each other. Several forms of dispute between the two are possible.

For example, suppose that John sends an authenticated message to Mary, using one of the schemes of Figure 12.1. Consider the following disputes that could arise.

1. Mary may forge a different message and claim that it came from John. Mary would simply have to create a message and append an authentication code using the key that John and Mary share.

2. John can deny sending the message. Because it is possible for Mary to forge a message, there is no way to prove that John did in fact send the message.

Digitalsignature

generationalgorithm

Bob’sprivate

key

Bob’spublic

key

Bob’ssignature

for M

S

Mes

sage

M

Digitalsignature

verificationalgorithm

Returnsignature

valid or not valid

Bob Transmit Alice

S

Mes

sage

M

Figure 13.1 Generic Model of Digital Signature Process


Both scenarios are of legitimate concern. Here is an example of the first scenario: An electronic funds transfer takes place, and the receiver increases the amount of funds transferred and claims that the larger amount had arrived from the sender. An example of the second scenario is that an electronic mail message contains instructions to a stockbroker for a transaction that subsequently turns out badly. The sender pretends that the message was never sent.

In situations where there is not complete trust between sender and receiver, something more than authentication is needed. The most attractive solution to this problem is the digital signature. The digital signature must have the following properties:

• It must verify the author and the date and time of the signature.

• It must authenticate the contents at the time of the signature.

• It must be verifiable by third parties, to resolve disputes.

Thus, the digital signature function includes the authentication function.

Bob Alice

EncryptCompare

Bob’ssignature

for M

S

Message M

Cryptographichash

function

h

Message M

Cryptographichash

function

h

Decrypt

h’

S

Returnsignature

valid or not valid

Bob’sprivate

key

Bob’spublic

key

Figure 13.2 Simplified Depiction of Essential Elements of Digital Signature Process

13.1 / Digital SignatureS 397

Attacks and Forgeries

[GOLD88] lists the following types of attacks, in order of increasing severity. Here A denotes the user whose signature method is being attacked, and C denotes the attacker.

• Key-only attack: C only knows A’s public key.

• Known message attack: C is given access to a set of messages and their signatures.

• Generic chosen message attack: C chooses a list of messages before attempt-ing to breaks A’s signature scheme, independent of A’s public key. C then obtains from A valid signatures for the chosen messages. The attack is generic, because it does not depend on A’s public key; the same attack is used against everyone.

• Directed chosen message attack: Similar to the generic attack, except that the list of messages to be signed is chosen after C knows A’s public key but before any signatures are seen.

• Adaptive chosen message attack: C is allowed to use A as an “oracle.” This means that C may request from A signatures of messages that depend on pre-viously obtained message-signature pairs.

[GOLD88] then defines success at breaking a signature scheme as an outcome in which C can do any of the following with a non-negligible probability:

• Total break: C determines A’s private key.

• Universal forgery: C finds an efficient signing algorithm that provides an equivalent way of constructing signatures on arbitrary messages.

• Selective forgery: C forges a signature for a particular message chosen by C.

• Existential forgery: C forges a signature for at least one message. C has no control over the message. Consequently, this forgery may only be a minor nui-sance to A.

Digital Signature Requirements

On the basis of the properties and attacks just discussed, we can formulate the fol-lowing requirements for a digital signature.

• The signature must be a bit pattern that depends on the message being signed.

• The signature must use some information unique to the sender to prevent both forgery and denial.

• It must be relatively easy to produce the digital signature.

• It must be relatively easy to recognize and verify the digital signature.

• It must be computationally infeasible to forge a digital signature, either by constructing a new message for an existing digital signature or by constructing a fraudulent digital signature for a given message.

• It must be practical to retain a copy of the digital signature in storage.


A secure hash function, embedded in a scheme such as that of Figure 13.2, provides a basis for satisfying these requirements. However, care must be taken in the design of the details of the scheme.

Direct Digital Signature

The term direct digital signature refers to a digital signature scheme that involves only the communicating parties (source, destination). It is assumed that the destina-tion knows the public key of the source.

Confidentiality can be provided by encrypting the entire message plus signa-ture with a shared secret key (symmetric encryption). Note that it is important to perform the signature function first and then an outer confidentiality function. In case of dispute, some third party must view the message and its signature. If the sig-nature is calculated on an encrypted message, then the third party also needs access to the decryption key to read the original message. However, if the signature is the inner operation, then the recipient can store the plaintext message and its signature for later use in dispute resolution.

The validity of the scheme just described depends on the security of the sender’s private key. If a sender later wishes to deny sending a particular message, the sender can claim that the private key was lost or stolen and that someone else forged his or her signature. Administrative controls relating to the security of private keys can be em-ployed to thwart or at least weaken this ploy, but the threat is still there, at least to some degree. One example is to require every signed message to include a timestamp (date and time) and to require prompt reporting of compromised keys to a central authority.

Another threat is that some private key might actually be stolen from X at time T. The opponent can then send a message signed with X’s signature and stamped with a time before or equal to T.

The universally accepted technique for dealing with these threats is the use of a digital certificate and certificate authorities. We defer a discussion of this topic until Chapter 14, and focus in this chapter on digital signature algorithms.

13.2 eLgamaL DigitaL signature scheme

Before examining the NIST Digital Signature Algorithm, it will be helpful to un-derstand the Elgamal and Schnorr signature schemes. Recall from Chapter 10, that the Elgamal encryption scheme is designed to enable encryption by a user’s public key with decryption by the user’s private key. The Elgamal signature scheme in-volves the use of the private key for encryption and the public key for decryption [ELGA84, ELGA85].

Before proceeding, we need a result from number theory. Recall from Chapter 8 that for a prime number q, if a is a primitive root of q, then

a, a2, c, aq - 1

are distinct (mod q). It can be shown that, if a is a primitive root of q, then

1. For any integer m, am K 1 (mod q) if and only if m K 0 (mod q - 1).

2. For any integers, i, j, ai K aj (mod q) if and only if i K j (mod q - 1).

13.2 / elgamal Digital Signature SCheme 399

As with Elgamal encryption, the global elements of Elgamal digital signature are a prime number q and a, which is a primitive root of q. User A generates a private/public key pair as follows.

1. Generate a random integer XA, such that 1 6 XA 6 q - 1.

2. Compute YA = aXA mod q.

3. A’s private key is XA; A’s pubic key is {q, a, YA}.

To sign a message M, user A first computes the hash m = H(M), such that m is an integer in the range 0 … m … q - 1. A then forms a digital signature as follows.

1. Choose a random integer K such that 1 … K … q - 1 and gcd(K, q - 1) = 1. That is, K is relatively prime to q - 1.

2. Compute S1 = aK mod q. Note that this is the same as the computation of C1 for Elgamal encryption.

3. Compute K - 1mod (q - 1). That is, compute the inverse of K modulo q - 1.

4. Compute S2 = K - 1(m - XAS1) mod (q - 1).

5. The signature consists of the pair (S1, S2).

Any user B can verify the signature as follows.

1. Compute V1 = am mod q.

2. Compute V2 = (YA)S1(S1)S2 mod q.

The signature is valid if V1 = V2. Let us demonstrate that this is so. Assume that the equality is true. Then we have

am mod q = (YA)S1(S1)S2 mod q assume V1 = V2

am mod q = aXAS1aKS2 mod q substituting for YA and S1

am - XA S1 mod q = aKS2 mod q rearranging termsm - XAS1 K KS2 mod (q - 1) property of primitive rootsm - XAS1 K KK - 1 (m - XAS1) mod (q - 1) substituting for S2

For example, let us start with the prime field GF(19); that is, q = 19. It has primitive roots {2, 3, 10, 13, 14, 15}, as shown in Table 8.3. We choose a = 10.

Alice generates a key pair as follows:

1. Alice chooses XA = 16.

2. Then YA = aXA mod q = a16 mod 19 = 4.

3. Alice’s private key is 16; Alice’s pubic key is {q, a, YA} = {19, 10, 4}.

Suppose Alice wants to sign a message with hash value m = 14.

1. Alice chooses K = 5, which is relatively prime to q - 1 = 18.

2. S1 = aK mod q = 105 mod 19 = 3 (see Table 8.3).

3. K - 1 mod (q - 1) = 5- 1 mod 18 = 11.


4. S2 = K - 1 (m - XAS1) mod (q - 1) = 11 (14 - (16)(3)) mod 18 = -374 mod 18 = 4.

Bob can verify the signature as follows.

1. V1 = am mod q = 1014 mod 19 = 16.

2. V2 = (YA)S1(S1)S2 mod q = (43)(34) mod 19 = 5184 mod 19 = 16.

Thus, the signature is valid.

13.3 schnOrr DigitaL signature scheme

As with the Elgamal digital signature scheme, the Schnorr signature scheme is based on discrete logarithms [SCHN89, SCHN91]. The Schnorr scheme minimizes the message-dependent amount of computation required to generate a signature. The main work for signature generation does not depend on the message and can be done during the idle time of the processor. The message-dependent part of the signature generation requires multiplying a 2n-bit integer with an n-bit integer.

The scheme is based on using a prime modulus p, with p - 1 having a prime factor q of appropriate size; that is, p - 1 K (mod q). Typically, we use p ≈ 21024 and q ≈ 2160. Thus, p is a 1024-bit number, and q is a 160-bit number, which is also the length of the SHA-1 hash value.

The first part of this scheme is the generation of a private/public key pair, which consists of the following steps.

1. Choose primes p and q, such that q is a prime factor of p - 1.

2. Choose an integer a, such that aq = 1 mod p. The values a, p, and q comprise a global public key that can be common to a group of users.

3. Choose a random integer s with 0 6 s 6 q. This is the user’s private key.

4. Calculate v = a- s mod p. This is the user’s public key.

A user with private key s and public key v generates a signature as follows.

1. Choose a random integer r with 0 6 r 6 q and compute x = ar mod p. This computation is a preprocessing stage independent of the message M to be signed.

2. Concatenate the message with x and hash the result to compute the value e:

e = H(M }x)

3. Compute y = (r + se) mod q. The signature consists of the pair (e, y).

Any other user can verify the signature as follows.

1. Compute x′ = ayve mod p.

2. Verify that e = H(M }x′).

To see that the verification works, observe that

x′ K ayve K aya- se K ay - se K ar K x (mod p)

Hence, H(M }x′) = H(M }x).

13.4 / niSt Digital Signature algorithm 401

13.4 nist DigitaL signature aLgOrithm

The National Institute of Standards and Technology (NIST) has published Federal Information Processing Standard FIPS 186, known as the Digital Signature Algorithm (DSA). The DSA makes use of the Secure Hash Algorithm (SHA) described in Chapter 12. The DSA was originally proposed in 1991 and revised in 1993 in response to public feedback concerning the se-curity of the scheme. There was a further minor revision in 1996. In 2000, an expanded version of the standard was issued as FIPS 186-2, subsequently up-dated to FIPS 186-3 in 2009. This latest version also incorporates digital sig-nature algorithms based on RSA and on elliptic curve cryptography. In this section, we discuss DSA.

The DSA Approach

The DSA uses an algorithm that is designed to provide only the digital signa-ture function. Unlike RSA, it cannot be used for encryption or key exchange. Nevertheless, it is a public-key technique.

Figure 13.3 contrasts the DSA approach for generating digital signatures to that used with RSA. In the RSA approach, the message to be signed is input to a hash function that produces a secure hash code of fixed length. This hash code is then encrypted using the sender’s private key to form the signature. Both the mes-sage and the signature are then transmitted. The recipient takes the message and produces a hash code. The recipient also decrypts the signature using the sender’s public key. If the calculated hash code matches the decrypted signature, the signa-ture is accepted as valid. Because only the sender knows the private key, only the sender could have produced a valid signature.

(a) RSA approach

M

H

| | M

Sig Ver

H

Compare

k

sr

(b) DSA approach

M

H

| | M

E D

H

ComparePRa

PRaPUG PUaPUG

PUa

E(PRa, H(M)]

Figure 13.3 Two Approaches to Digital Signatures


The DSA approach also makes use of a hash function. The hash code is pro-vided as input to a signature function along with a random number k generated for this particular signature. The signature function also depends on the sender’s private key (PRa) and a set of parameters known to a group of communicating prin-cipals. We can consider this set to constitute a global public key (PUG).1 The result is a signature consisting of two components, labeled s and r.

At the receiving end, the hash code of the incoming message is generated. This plus the signature is input to a verification function. The verification function also depends on the global public key as well as the sender’s public key (PUa), which is paired with the sender’s private key. The output of the verification function is a value that is equal to the signature component r if the signature is valid. The signa-ture function is such that only the sender, with knowledge of the private key, could have produced the valid signature.

We turn now to the details of the algorithm.

The Digital Signature Algorithm

The DSA is based on the difficulty of computing discrete logarithms (see Chapter 8) and is based on schemes originally presented by Elgamal [ELGA85] and Schnorr [SCHN91].

Figure 13.4 summarizes the algorithm. There are three parameters that are public and can be common to a group of users. An N-bit prime number q is chosen. Next, a prime number p is selected with a length between 512 and 1024 bits such that q divides (p - 1). Finally, g is chosen to be of the form h(p - 1)/q mod p, where h is an integer between 1 and (p - 1) with the restriction that g must be greater than 1.2 Thus, the global public-key components of DSA are the same as in the Schnorr signature scheme.

With these numbers in hand, each user selects a private key and generates a public key. The private key x must be a number from 1 to (q - 1) and should be chosen randomly or pseudorandomly. The public key is calculated from the pri-vate key as y = gx mod p. The calculation of y given x is relatively straightforward. However, given the public key y, it is believed to be computationally infeasible to de-termine x, which is the discrete logarithm of y to the base g, mod p (see Chapter 8).

To create a signature, a user calculates two quantities, r and s, that are func-tions of the public key components (p, q, g), the user’s private key (x), the hash code of the message H(M), and an additional integer k that should be generated randomly or pseudorandomly and be unique for each signing.

Let M, r′, and s′ be the received versions of M, r, and s, respectively. Verification is performed using the formulas shown in Figure 13.4. The receiver generates a quantity v that is a function of the public key components, the sender’s public key, and the hash code of the incoming message. If this quantity matches the r component of the signature, then the signature is validated.

2In number-theoretic terms, g is of order q mod p; see Chapter 8.

1It is also possible to allow these additional parameters to vary with each user so that they are a part of a user’s public key. In practice, it is more likely that a global public key will be used that is separate from each user’s public key.

13.4 / niSt Digital Signature algorithm 403

Figure 13.5 depicts the functions of signing and verifying.The structure of the algorithm, as revealed in Figure 13.5, is quite interesting.

Note that the test at the end is on the value r, which does not depend on the message at all. Instead, r is a function of k and the three global public-key components. The multiplicative inverse of k (mod q) is passed to a function that also has as inputs the message hash code and the user’s private key. The structure of this function is such that the receiver can recover r using the incoming message and signature, the public key of the user, and the global public key. It is certainly not obvious from Figure 13.4 or Figure 13.5 that such a scheme would work. A proof is provided in Appendix K.

Given the difficulty of taking discrete logarithms, it is infeasible for an oppo-nent to recover k from r or to recover x from s.

Another point worth noting is that the only computationally demanding task in signature generation is the exponential calculation gk mod p. Because this value does not depend on the message to be signed, it can be computed ahead of time. Indeed, a user could precalculate a number of values of r to be used to sign documents as needed. The only other somewhat demanding task is the determi-nation of a multiplicative inverse, k-1. Again, a number of these values can be precalculated.

Global Public-Key Components

p prime number where 2L - 1 6 p 6 2L for 512 … L … 1024 and L a multiple of 64; i.e., bit length of between 512 and 1024 bits in increments of 64 bits

q prime divisor of ( p - 1), where 2N - 1 6 q 6 2N i.e., bit length of N bits

g = h( p - 1)/q mod p, where h is any integer with 1 6 h 6 ( p - 1) such that h( p - 1)/q mod p 7 1

User’s Private Key

x random or pseudorandom integer with 0 6 x 6 q

User’s Public Key

y = gx mod p

User’s Per-Message Secret Number

k random or pseudorandom integer with 0 6 k 6 q

Signing

r = (gk mod p) mod q

s = [k-1 (H(M) + xr)] mod q

Signature = (r, s)

Verifying

w = (s′)-1 mod q

u1 = [H(M′)w] mod q

u2 = (r′)w mod q

v = [(gu1 yu2) mod p] mod q

TEST: v = r′

M = message to be signed

H(M) = hash of M using SHA-1

M′, r′, s′ = received versions of M, r, s

Figure 13.4 The Digital Signature Algorithm (DSA)


13.5 eLLiptic curve DigitaL signature aLgOrithm

As was mentioned, the 2009 version of FIPS 186 includes a new digital signature technique based on elliptic curve cryptography, known as the Elliptic Curve Digital Signature Algorithm (ECDSA). ECDSA is enjoying increasing acceptance due to the efficiency advantage of elliptic curve cryptography, which yields security com-parable to that of other schemes with a smaller key bit length.

First we give a brief overview of the process involved in ECDSA. In essence, four elements are involved.

1. All those participating in the digital signature scheme use the same global domain parameters, which define an elliptic curve and a point of origin on the curve.

(a) Signing

(b) Verifying

M

sr

M'

s'r'

H

r = (gk mod p) mod q

q

s = [k–1 (H(M) + xr)] mod q

w = (s')–1 mod q

k k

k

q

x x

M

H(M)

HH(M')

p g

q

q

y

v

g

u1 = [H(M')w)] mod qu2 = (r')w mod qv = [(gu1yu2) mod p] mod q

signatureverification

r' = v?

r'

r'

w

Figure 13.5 DSA Signing and Verifying

13.5 / elliptiC Curve Digital Signature algorithm 405

2. A signer must first generate a public, private key pair. For the private key, the signer selects a random or pseudorandom number. Using that random number and the point of origin, the signer computes another point on the elliptic curve. This is the signer’s public key.

3. A hash value is generated for the message to be signed. Using the private key, the domain parameters, and the hash value, a signature is generated. The signa-ture consists of two integers, r and s.

4. To verify the signature, the verifier uses as input the signer’s public key, the domain parameters, and the integer s. The output is a value v that is compared to r. The signature is verified if the v = r.

Let us examine each of these four elements in turn.

Global Domain Parameters

Recall from Chapter 10 that two families of elliptic curves are used in cryptographic applications: prime curves over Zp and binary curves over GF(2m). For ECDSA, prime curves are used. The global domain parameters for ECDSA are the following:

q a prime number

a, b integers that specify the elliptic curve equation defined over Zq with the equation y2 = x3 + ax + b

G a base point represented by G = (xg, yg) on the elliptic curve equationn order of point G; that is, n is the smallest positive integer such that

nG = O. This is also the number of points on the curve.

Key Generation

Each signer must generate a pair of keys, one private and one public. The signer, let us call him Bob, generates the two keys using the following steps:

1. Select a random integer d, d ∈ [1, n - 1]

2. Compute Q = dG. This is a point in Eq(a, b).

3. Bob’s public key is Q and private key is d.

Digital Signature Generation and Authentication

With the public domain parameters and a private key in hand, Bob generates a digi-tal signature of 320 bytes for message m using the following steps:

1. Select a random or pseudorandom integer k, k ∈ [1, n - 1]

2. Compute point P = (x, y) = kG and r = x mod n. If r = 0 then goto step 1

3. Compute t = k-1 mod n

4. Compute e = H(m), where H is the SHA-1 hash function, which produces a 160-bit hash value

5. Compute s = k-1(e + dr) mod n. If s = O then goto step 1

6. The signature of message m is the pair (r, s).


Alice knows the public domain parameters and Bob’s public key. Alice is pre-sented with Bob’s message and digital signature and verifies the signature using the following steps:

1. Verify that r and s are integers in the range 1 through n - 1 2. Using SHA-1, compute the 160-bit hash value e = H(m) 3. Compute w = s-1 mod n 4. Compute u1 = ew and u2 = rw

5. Compute the point X = (x1, y1) = u1G + u2Q

6. If X = O, reject the signature else compute v = x1 mod n 7. Accept Bob’s signature if and only if v = r

Figure 13.6 illustrates the signature authentication process. We can verify that this process is valid as follows. If the message received by Alice is in fact signed by Bob, then

s = k-1(e + dr) mod n

Q

Yes

No

No

No

No

No

Yes

Yes

Yes

Yes

r, s

Acceptsignature

Rejectsignature

Bob Alice

Generate k(x, y) = kGr = x mod n

Generate privatekey d. Publickey Q = dG

q, a, b, G, nare sharedglobal variables

Signature of mis r, s

v = x1 mod n

e = H(m)s = k–1 (e + dr) mod n

e = H(m)w = s–1 mod nu1 = ew, u2 = rwX = (x1, x2) = u1G + u2Q

r = 0?

s = 0?

X = O?

v = r?

r, s integersin range [1, n–1]?

Figure 13.6 ECDSA Signing and Verifying

13.6 / rSa-pSS Digital Signature algorithm 407

Then

k = s-1(e + dr) mod n

k = (s-1e + s-1dr) mod n

k = (we + wdr) mod n

k = (u1 + u2d) mod n

Now consider that

u1G + u2Q = u1G + u2dG = (u1 + u2d)G = kG

In step 5 of the verification process, we have v = x1 mod n, where point X = (x1, y1) = u1G + u2Q. Thus we see that v = r since r = x mod n and x is the x coor-dinate of the point kG and we have already seen that u1G + u2Q = kG.

13.6 rsa-pss DigitaL signature aLgOrithm

In addition to the NIST Digital Signature Algorithm and ECDSA, the 2009 ver-sion of FIPS 186 also includes several techniques based on RSA, all of which were developed by RSA Laboratories and are in wide use. In this section, we discuss the RSA Probabilistic Signature Scheme (RSA-PSS), which is the latest of the RSA schemes and the one that RSA Laboratories recommends as the most secure of the RSA schemes.

Because the RSA-based schemes are widely deployed in many applications, including financial applications, there has been great interest in demonstrating that such schemes are secure. The three main RSA signature schemes differ mainly in the padding format the signature generation operation employs to embed the hash value into a message representative, and in how the signature verification opera-tion determines that the hash value and the message representative are consistent. For all of the schemes developed prior to PSS, it has not been possible to develop a mathematical proof that the signature scheme is as secure as the underlying RSA encryption/decryption primitive [KALI01]. The PSS approach was first proposed by Bellare and Rogaway [BELL96c, BELL98]. This approach, unlike the other RSA-based schemes, introduces a randomization process that enables the security of the method to be shown to be closely related to the security of the RSA algorithm itself. This makes RSA-PSS more desirable as the choice for RSA-based digital sig-nature applications.

Mask Generation Function

Before explaining the RSA-PSS operation, we need to describe the mask generation function (MGF) used as a building block. MGF(X, maskLen) is a pseudorandom function that has as input parameters a bit string X of any length and the desired length L in octets of the output. MGFs are typically based on a secure cryptographic hash function such as SHA-1. An MGF based on a hash function is intended to be a cryptographically secure way of generating a message digest, or hash, of variable length based on an underlying cryptographic hash function that produces a fixed-length output.


The MGF function used in the current specification for RSA-PSS is MGF1, with the following parameters:

Options Hash hash function with output hLen octets

Input X octet string to be masked

maskLen length in octets of the mask

Output mask an octet string of length maskLen

MGF1 is defined as follows:


T = empty string

k = <maskLen/hLen= − 1 2. Calculate intermediate values.

for counter = 0 to k

Represent counter as a 32-bit string C

T = T } Hash(X } C)

3. Output results.

mask = the leading maskLen octets of T

In essence, MGF1 does the following. If the length of the desired output is equal to the length of the hash value (maskLen = hLen), then the output is the hash of the input value X concatenated with a 32-bit counter value of 0. If maskLen is greater than hLen, the MGF1 keeps iterating by hashing X concatenated with the counter and appending that to the current string T. So that the output is

Hash(X }0) }Hash(X }1) } c}Hash(X }k)

This is repeated until the length of T is greater than or equal to maskLen, at which point the output is the first maskLen octets of T.

The Signing Operation

Message encoding The first stage in generating an RSA-PSS signature of a mes-sage M is to generate from M a fixed-length message digest, called an encoded mes-sage (EM). Figure 13.7 illustrates this process. We define the following parameters and functions:

Options Hash hash function with output hLen octets. The current pre-ferred alternative is SHA-1, which produces a 20-octet hash value.

MGF mask generation function. The current specification calls for MGF1.

sLen length in octets of the salt. Typically sLen = hLen, which for the current version is 20 octets.


Input M message to be encoded for signing.emBits This value is one less than the length in bits of the RSA

modulus n.Output EM encoded message. This is the message digest that will be

encrypted to form the digital signature.Parameters emLen length of EM in octets = <emBits/8= .

padding1 hexadecimal string 00 00 00 00 00 00 00 00; that is, a string of 64 zero bits.

padding2 hexadecimal string of 00 octets with a length (emLen - sLen - hLen - 2) octets, followed by the hexadecimal octet with value 01.

salt a pseudorandom number.bc the hexadecimal value BC.

The encoding process consists of the following steps.

1. Generate the hash value of M: mHash = Hash(M)

2. Generate a pseudorandom octet string salt and form block M¿ = padding1 ‘ mHash ‘ salt

3. Generate the hash value of M¿: H = Hash(M¿) 4. Form data block DB = padding2 ‘ salt

5. Calculate the MGF value of H: dbMask = MGF(H, emLen - hLen - 1)

6. Calculate maskedDB = DB ⊕ dbMsk

Hash

Hash

MGF

M

mHash saltpadding1

bcmaskedDB

saltpadding2

M' =

DB =

EM = H

Figure 13.7 RSA-PSS Encoding


7. Set the leftmost 8emLen - emBits bits of the leftmost octet in maskedDB to 0

8. EM = maskedDB ‘ H ‘ bc

We make several comments about the complex nature of this message digest algorithm. All of the RSA-based standardized digital signature schemes involve ap-pending one or more constants (e.g., padding1 and padding2) in the process of form-ing the message digest. The objective is to make it more difficult for an adversary to find another message that maps to the same message digest as a given message or to find two messages that map to the same message digest. RSA-PSS also incorporates a pseudorandom number, namely the salt. Because the salt changes with every use, signing the same message twice using the same private key will yield two different signatures. This is an added measure of security.

ForMing the signature We now show how the signature is formed by a signer with private key {d, n} and public key {e, n} (see Figure 9.5). Treat the octet string EM as an unsigned, nonnegative binary integer m. The signature s is formed by en-crypting m as follows:

s = md mod n

Let k be the length in octets of the RSA modulus n. For example if the key size for RSA is 2048 bits, then k = 2048/8 = 256. Then convert the signature value s into the octet string S of length k octets.

Signature Verification

decryption For signature verification, treat the signature S as an unsigned, nonneg-ative binary integer s. The message digest m is recovered by decrypting s as follows:

m = se mod n

Then, convert the message representative m to an encoded message EM of length emLen = <modBits - 1)/8= octets, where modBits is the length in bits of the RSA modulus n.

eM VeriFication EM verification can be described as follows:

Options Hash hash function with output hLen octets.

MGF mask generation function.

sLen length in octets of the salt.

Input M message to be verified.

EM the octet string representing the decrypted signature, with length emLen = <emBits/8= .

emBits This value is one less than the length in bits of the RSA modulus n.

Parameters padding1 hexadecimal string 00 00 00 00 00 00 00 00; that is, a string of 64 zero bits.

padding2 Hexadecimal string of 00 octets with a length (emLen - sLen - hLen - 2) octets, followed by the hexadecimal octet with value 01.


1. Generate the hash value of M: mHash = Hash(M)

2. If emLen 6 hLen + sLen + 2, output “inconsistent” and stop

3. If the rightmost octet of EM does not have hexadecimal value BC, output “inconsistent” and stop

4. Let maskedDB be the leftmost emLen - hLen - 1 octets of EM, and let H be the next hLen octets

5. If the leftmost 8emLen - emBits bits of the leftmost octet in maskedDB are not all equal to zero, output “inconsistent” and stop

6. Calculate dbMask = MGF (H, emLen - hLen - 1)

7. Calculate DB = maskedDB ⊕ dbMsk

8. Set the leftmost 8emLen − emBits bits of the leftmost octet in DB to zero

9. If the leftmost (emLen - hLen - sLen - 1) octets of DB are not equal to padding2, output “inconsistent” and stop

10. Let salt be the last sLen octets of DB

11. Form block M¿ = padding1 ‘ mHash ‘ salt

12. Generate the hash value of M¿: H¿ = Hash(M¿) 13. If H = H¿, output “consistent.” Otherwise, output “inconsistent”

Figure 13.8 illustrates the process. The shaded boxes labeled H and H¿ corre-spond, respectively, to the value contained in the decrypted signature and the value

Hash

Hash

MGF

M

mHash saltpadding1

maskedDB

dbMask

salt

= M'

DB =

EM = H

H'

Figure 13.8 RSA-PSS EM Verification


generated from the message M associated with the signature. The remaining three shaded areas contain values generated from the decrypted signature and compared to known constants. We can now see more clearly the different roles played by the constants and the pseudorandom value salt, all of which are embedded in the EM generated by the signer. The constants are known to the verifier, so that the com-puted constants can be compared to the known constants as an additional check that the signature is valid (in addition to comparing H and H¿). The salt results in a different signature every time a given message is signed with the same private key. The verifier does not know the value of the salt and does not attempt a comparison. Thus, the salt plays a similar role to the pseudorandom variable k in the NIST DSA and in ECDSA. In both of those schemes, k is a pseudorandom number generated by the signer, resulting in different signatures from multiple signings of the same message with the same private key. A verifier does not and need not know the value of k.

AKL83 Akl, S. “Digital Signatures: A Tutorial Survey.” Computer, February 1983.MITC92 Mitchell, C.; Piper, F. ; and Wild, P. “Digital Signatures.” In [SIMM92].


Key Terms

direct digital signaturedigital signatureDigital Signature Algorithm

(DSA)

Elgamal digital signatureElliptic Curve Digital

Signature Algorithm (ECDSA)

Schnorr digital signaturetimestamp

Review Questions 13.1 List two disputes that can arise in the context of message authentication. 13.2 What are the properties a digital signature should have? 13.3 What requirements should a digital signature scheme satisfy? 13.4 What is the difference between direct and arbitrated digital signature? 13.5 In what order should the signature function and the confidentiality function be ap-

plied to a message, and why? 13.6 What are some threats associated with a direct digital signature scheme?


[AKL83] is the classic paper on digital signatures and is still highly relevant. Another excel-lent survey is [MITC92].

13.8 / Key termS, review QueStionS, anD problemS 413

Problems 13.1 Dr. Watson patiently waited until Sherlock Holmes finished. “Some interesting prob-

lem to solve, Holmes?” he asked when Holmes finally logged out.“Oh, not exactly. I merely checked my e-mail and then made a couple of net-

work experiments instead of my usual chemical ones. I have only one client now and I have already solved his problem. If I remember correctly, you once mentioned cryp-tology among your other hobbies, so it may interest you.”

“Well, I am only an amateur cryptologist, Holmes. But of course I am inter-ested in the problem. What is it about?”

“My client is Mr. Hosgrave, director of a small but progressive bank. The bank is fully computerized and of course uses network communications extensively. The bank already uses RSA to protect its data and to digitally sign documents that are communicated. Now the bank wants to introduce some changes in its procedures; in particular, it needs to digitally sign some documents by two signatories.

1. The first signatory prepares the document, forms its signature, and passes the doc-ument to the second signatory.

2. The second signatory as a first step must verify that the document was really signed by the first signatory. She then incorporates her signature into the docu-ment’s signature so that the recipient, as well as any member of the public, may verify that the document was indeed signed by both signatories. In addition, only the second signatory has to be able to verify the document’s signature after the first step; that is, the recipient (or any member of the public) should be able to verify only the complete document with signatures of both signatories, but not the document in its intermediate form where only one signatory has signed it. Moreover, the bank would like to make use of its existing modules that support RSA-style digital signatures.”

“Hm, I understand how RSA can be used to digitally sign documents by one signa-tory, Holmes. I guess you have solved the problem of Mr. Hosgrave by appropriate generalization of RSA digital signatures.”

“Exactly, Watson,” nodded Sherlock Holmes. “Originally, the RSA digital sig-nature was formed by encrypting the document by the signatory’s private decryption key ‘d’, and the signature could be verified by anyone through its decryption using publicly known encryption key ‘e’. One can verify that the signature S was formed by the person who knows d, which is supposed to be the only signatory. Now the problem of Mr. Hosgrave can be solved in the same way by slight generalization of the process, that is …”

Finish the explanation. 13.2 DSA specifies that if the signature generation process results in a value of s = 0,

a new value of k should be generated and the signature should be recalculated. Why?

13.3 What happens if a k value used in creating a DSA signature is compromised? 13.4 The DSA document includes a recommended algorithm for testing a number for

primality. 1. [Choose w] Let w be a random odd integer. Then (w - 1) is even and can be

expressed in the form 2am with m odd. That is, 2a is the largest power of 2 that divides (w - 1).

2. [Generate b] Let b be a random integer in the range 1 6 b 6 w. 3. [Exponentiate] Set j = 0 and z = bm mod w. 4. [Done?] If j = 0 and z = 1, or if z = w - 1, then w passes the test and may be

prime; go to step 8. 5. [Terminate?] If j 7 0 and z = 1, then w is not prime; terminate algorithm for

this w. 6. [Increase j] Set j = j + 1. If j 6 a, set z = z2mod w and go to step 4. 7. [Terminate] w is not prime; terminate algorithm for this w.


8. [Test again?] If enough random values of b have been tested, then accept w as prime and terminate algorithm; otherwise, go to step 2.

a. Explain how the algorithm works. b. Show that it is equivalent to the Miller-Rabin test described in Chapter 8.

13.5 With DSA, because the value of k is generated for each signature, even if the same message is signed twice on different occasions, the signatures will differ. This is not true of RSA signatures. What is the practical implication of this difference?

13.6 Consider the problem of creating domain parameters for DSA. Suppose we have already found primes p and q such that q|(p - 1). Now we need to find g ∈ Zp with g of order q mod p. Consider the following two algorithms:

Algorithm 1 Algorithm 2

repeat repeatselect g ∈ Zp select h ∈ Zp

h d gq mod p g d h(p - l)/q mod puntil (h = 1 and g ≠ 1) until (g ≠ 1)return g return g

a. Prove that the value returned by Algorithm 1 has order q. b. Prove that the value returned by Algorithm 2 has order q. c. Suppose p = 40193 and q = 157. How many loop iterations do you expect

Algorithm 1 to make before it finds a generator? d. If p is 1024 bits and q is 160 bits, would you recommend using Algorithm 1 to

find g? Explain. e. Suppose p = 40193 and q = 157. What is the probability that Algorithm 2 com-

putes a generator in its very first loop iteration? (If it is helpful, you may use the

fact that ad �n

w(d) = n when answering this question.)

13.7 It is tempting to try to develop a variation on Diffie-Hellman that could be used as a digital signature. Here is one that is simpler than DSA and that does not require a secret random number in addition to the private key.

Public elements: q prime numbera a 6 q and a is a primitive root of q

Private key: X X 6 q

Public key: Y = aX mod q

To sign a message M, compute h = H(M), which is the hash code of the message. We require that gcd(h, q - 1) = 1. If not, append the hash to the message and calculate a new hash. Continue this process until a hash code is produced that is relatively prime to (q - 1). Then calculate Z to satisfy Z * h K X(mod q - 1). The signature of the message is aZ. To verify the signature, a user verifies that Y = (aZ)h = aXmod q.

a. Show that this scheme works. That is, show that the verification process produces an equality if the signature is valid.

b. Show that the scheme is unacceptable by describing a simple technique for forging a user’s signature on an arbitrary message.

13.8 An early proposal for a digital signature scheme using symmetric encryption is based on the following. To sign an n-bit message, the sender randomly generates in advance 2n 56-bit cryptographic keys:

k1, K1, k2, K2, c, kn, Kn

13.8 / Key termS, review QueStionS, anD problemS 415

which are kept private. The sender prepares in advance two sets of corresponding non-secret 64-bit validation parameters, which are made public:

u1, U1, u2, U2, c, un, Un and v1, V1, v2, V2, c, vn, Vn

where

vi = E(ki, ui), Vi = E(ki, Ui)

The message M is signed as follows. For the ith bit of the message, either ki or Ki is attached to the message, depending on whether the message bit is 0 or 1. For example, if the first three bits of the message are 011, then the first three keys of the signature are k1, K2, K3.

a. How does the receiver validate the message? b. Is the technique secure? c. How many times can the same set of secret keys be safely used for different

messages? d. What, if any, practical problems does this scheme present?


417

14.1 Symmetric Key Distribution Using Symmetric Encryption

A Key Distribution ScenarioHierarchical Key ControlSession Key LifetimeA Transparent Key Control SchemeDecentralized Key ControlControlling Key Usage

14.2 Symmetric Key Distribution Using Asymmetric Encryption

Simple Secret Key DistributionSecret Key Distribution with Confidentiality and AuthenticationA Hybrid Scheme

14.3 Distribution of Public Keys

Public Announcement of Public KeysPublicly Available DirectoryPublic-Key AuthorityPublic-Key Certificates

14.4 X.509 Certificates

CertificatesX.509 Version 3

14.5 Public-Key Infrastructure

PKIX Management FunctionsPKIX Management Protocols



Part 4: Mutual trust

Chapter

Key ManageMent and distribution

417

418 Chapter 14 / Key ManageMent and distribution

No Singhalese, whether man or woman, would venture out of the house without a bunch of keys in his hand, for without such a talisman he would fear that some devil might take advantage of his weak state to slip into his body.


“Suppose that Cadogan West wished to make his way into the building after hours; he would need three keys, would he not, before the could reach the papers?”

“Yes, he would. The key of the outer door, the key of the office, and the key of the safe.”

—The Adventure of the Bruce-Partington Plans, Sir Arthur Conan Doyle

The topics of cryptographic key management and cryptographic key distribution are complex, involving cryptographic, protocol, and management considerations. The pur-pose of this chapter is to give the reader a feel for the issues involved and a broad sur-vey of the various aspects of key management and distribution. For more information, the place to start is the three-volume NIST SP 800-57, followed by the recommended readings listed at the end of this chapter.

14.1 Symmetric Key DiStribution uSing Symmetric encryption

For symmetric encryption to work, the two parties to an exchange must share the same key, and that key must be protected from access by others. Furthermore, fre-quent key changes are usually desirable to limit the amount of data compromised if an attacker learns the key. Therefore, the strength of any cryptographic system rests with the key distribution technique, a term that refers to the means of delivering a key to two parties who wish to exchange data without allowing others to see the key.

Learning objectiveS


u Discuss the concept of a key hierarchy.

u Understand the issues involved in using asymmetric encryption to distrib-ute symmetric keys.

u Present an overview of approaches to public-key distribution and analyze the risks involved in various approaches.

u List and explain the elements in an X.509 certificate.

u Present an overview of public-key infrastructure concepts.

14.1 / syMMetriC Key distribution using syMMetriC enCryption 419

For two parties A and B, key distribution can be achieved in a number of ways, as follows:

1. A can select a key and physically deliver it to B.

2. A third party can select the key and physically deliver it to A and B.

3. If A and B have previously and recently used a key, one party can transmit the new key to the other, encrypted using the old key.

4. If A and B each has an encrypted connection to a third party C, C can deliver a key on the encrypted links to A and B.

Options 1 and 2 call for manual delivery of a key. For link encryption, this is a reasonable requirement, because each link encryption device is going to be exchanging data only with its partner on the other end of the link. However, for end-to-end encryption over a network, manual delivery is awkward. In a distrib-uted system, any given host or terminal may need to engage in exchanges with many other hosts and terminals over time. Thus, each device needs a number of keys supplied dynamically. The problem is especially difficult in a wide-area dis-tributed system.

The scale of the problem depends on the number of communicating pairs that must be supported. If end-to-end encryption is done at a network or IP level, then a key is needed for each pair of hosts on the network that wish to communicate. Thus, if there are N hosts, the number of required keys is [N(N - 1)]/2. If encryp-tion is done at the application level, then a key is needed for every pair of users or processes that require communication. Thus, a network may have hundreds of hosts but thousands of users and processes. Figure 14.1 illustrates the magnitude of the key distribution task for end-to-end encryption.1 A network using node-level encryption with 1000 nodes would conceivably need to distribute as many as half a million keys. If that same network supported 10,000 applications, then as many as 50 million keys may be required for application-level encryption.

Returning to our list, option 3 is a possibility for either link encryption or end-to-end encryption, but if an attacker ever succeeds in gaining access to one key, then all subsequent keys will be revealed. Furthermore, the initial distribution of poten-tially millions of keys still must be made.

For end-to-end encryption, some variation on option 4 has been widely adopted. In this scheme, a key distribution center is responsible for distributing keys to pairs of users (hosts, processes, applications) as needed. Each user must share a unique key with the key distribution center for purposes of key distribution.

The use of a key distribution center is based on the use of a hierarchy of keys. At a minimum, two levels of keys are used (Figure 14.2). Communication between end systems is encrypted using a temporary key, often referred to as a session key. Typically, the session key is used for the duration of a logical connection, such as a frame relay connection or transport connection, and then discarded. Each session key is obtained from the key distribution center over the same networking facilities

1Note that this figure uses a log-log scale, so that a linear graph indicates exponential growth. A basic review of log scales is in the math refresher document at the Computer Science Student Resource Site at WilliamStallings.com/StudentSupport.html.


109

108

107

106

103 104 105

Num

ber

of k

eys

5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9

Number of endpoints

Figure 14.1 Number of Keys Required to Support Arbitrary Connections between Endpoints

Data Cryptographicprotection

Session keys Cryptographicprotection

Master keys Non-cryptographicprotection

Figure 14.2 The Use of a Key Hierarchy


used for end-user communication. Accordingly, session keys are transmitted in encrypted form, using a master key that is shared by the key distribution center and an end system or user.

For each end system or user, there is a unique master key that it shares with the key distribution center. Of course, these master keys must be distributed in some fashion. However, the scale of the problem is vastly reduced. If there are N entities that wish to communicate in pairs, then, as was mentioned, as many as [N(N - 1)]/2 session keys are needed at any one time. However, only N master keys are required, one for each entity. Thus, master keys can be distributed in some non-cryptographic way, such as physical delivery.

A Key Distribution Scenario

The key distribution concept can be deployed in a number of ways. A typical scenario is illustrated in Figure 14.3, which is based on a figure in [POPE79]. The scenario assumes that each user shares a unique master key with the key distribu-tion center (KDC).

Let us assume that user A wishes to establish a logical connection with B and requires a one-time session key to protect the data transmitted over the connection. A has a master key, Ka, known only to itself and the KDC; similarly, B shares the master key Kb with the KDC. The following steps occur.

1. A issues a request to the KDC for a session key to protect a logical connection to B. The message includes the identity of A and B and a unique identifier, N1, for this transaction, which we refer to as a nonce. The nonce may be a

Key DistributionCenter (KDC)

Keydistribution

steps

Authenticationsteps

Initiator A Responder B

(1) IDA || IDB || N1

(2) E(Ka, [Ks || IDA || IDB || N1])|| E(Kb, [Ks || IDA])

(3) E(Kb, [Ks || IDA])

(4) E(Ks, N2)

(5) E(Ks, f(N2))

Figure 14.3 Key Distribution Scenario


timestamp, a counter, or a random number; the minimum requirement is that it differs with each request. Also, to prevent masquerade, it should be difficult for an opponent to guess the nonce. Thus, a random number is a good choice for a nonce.

2. The KDC responds with a message encrypted using Ka. Thus, A is the only one who can successfully read the message, and A knows that it originated at the KDC. The message includes two items intended for A:

• The one-time session key, Ks, to be used for the session

• The original request message, including the nonce, to enable A to match this response with the appropriate request

Thus, A can verify that its original request was not altered before reception by the KDC and, because of the nonce, that this is not a replay of some previous request.

In addition, the message includes two items intended for B:

• The one-time session key, Ks, to be used for the session

• An identifier of A (e.g., its network address), IDA

These last two items are encrypted with Kb (the master key that the KDC shares with B). They are to be sent to B to establish the connection and prove A’s identity.

3. A stores the session key for use in the upcoming session and forwards to B the information that originated at the KDC for B, namely, E(Kb,[Ks } IDA]). Because this information is encrypted with Kb, it is protected from eaves-dropping. B now knows the session key (Ks), knows that the other party is A (from IDA), and knows that the information originated at the KDC (because it is encrypted using Kb).

At this point, a session key has been securely delivered to A and B, and they may begin their protected exchange. However, two additional steps are desirable:

4. Using the newly minted session key for encryption, B sends a nonce, N2, to A.

5. Also, using Ks, A responds with f(N2), where f is a function that performs some transformation on N2 (e.g., adding one).

These steps assure B that the original message it received (step 3) was not a replay.Note that the actual key distribution involves only steps 1 through 3, but that

steps 4 and 5, as well as step 3, perform an authentication function.

Hierarchical Key Control

It is not necessary to limit the key distribution function to a single KDC. Indeed, for very large networks, it may not be practical to do so. As an alternative, a hierarchy of KDCs can be established. For example, there can be local KDCs, each respon-sible for a small domain of the overall internetwork, such as a single LAN or a single building. For communication among entities within the same local domain, the local KDC is responsible for key distribution. If two entities in different domains desire a shared key, then the corresponding local KDCs can communicate through a global


KDC. In this case, any one of the three KDCs involved can actually select the key. The hierarchical concept can be extended to three or even more layers, depending on the size of the user population and the geographic scope of the internetwork.

A hierarchical scheme minimizes the effort involved in master key distribu-tion, because most master keys are those shared by a local KDC with its local enti-ties. Furthermore, such a scheme limits the damage of a faulty or subverted KDC to its local area only.

Session Key Lifetime

The more frequently session keys are exchanged, the more secure they are, because the opponent has less ciphertext to work with for any given session key. On the other hand, the distribution of session keys delays the start of any exchange and places a burden on network capacity. A security manager must try to balance these competing considerations in determining the lifetime of a particular session key.

For connection-oriented protocols, one obvious choice is to use the same ses-sion key for the length of time that the connection is open, using a new session key for each new session. If a logical connection has a very long lifetime, then it would be prudent to change the session key periodically, perhaps every time the PDU (protocol data unit) sequence number cycles.

For a connectionless protocol, such as a transaction-oriented protocol, there is no explicit connection initiation or termination. Thus, it is not obvious how often one needs to change the session key. The most secure approach is to use a new ses-sion key for each exchange. However, this negates one of the principal benefits of connectionless protocols, which is minimum overhead and delay for each transac-tion. A better strategy is to use a given session key for a certain fixed period only or for a certain number of transactions.

A Transparent Key Control Scheme

The approach suggested in Figure 14.3 has many variations, one of which is described in this subsection. The scheme (Figure 14.4) is useful for providing end-to-end encryption at a network or transport level in a way that is transparent to the end users. The approach assumes that communication makes use of a connec-tion-oriented end-to-end protocol, such as TCP. The noteworthy element of this approach is a session security module (SSM), which may consist of functionality at one protocol layer, that performs end-to-end encryption and obtains session keys on behalf of its host or terminal.

The steps involved in establishing a connection are shown in Figure 14.4. When one host wishes to set up a connection to another host, it transmits a connection- request packet (step 1). The SSM saves that packet and applies to the KDC for permis-sion to establish the connection (step 2). The communication between the SSM and the KDC is encrypted using a master key shared only by this SSM and the KDC. If the KDC approves the connection request, it generates the session key and delivers it to the two appropriate SSMs, using a unique permanent key for each SSM (step 3). The requesting SSM can now release the connection request packet, and a connection is set up between the two end systems (step 4). All user data exchanged between the two end systems are encrypted by their respective SSMs using the one-time session key.


The automated key distribution approach provides the flexibility and dynamic characteristics needed to allow a number of terminal users to access a number of hosts and for the hosts to exchange data with each other.

Decentralized Key Control

The use of a key distribution center imposes the requirement that the KDC be trusted and be protected from subversion. This requirement can be avoided if key distribu-tion is fully decentralized. Although full decentralization is not practical for larger networks using symmetric encryption only, it may be useful within a local context.

A decentralized approach requires that each end system be able to commu-nicate in a secure manner with all potential partner end systems for purposes of

Keydistribution

center

Network

1. Host sends packet requesting connection.2. Security service buffers packet; asks KDC for session key.3. KDC distributes session key to both hosts.4. Buffered packet transmitted.

HOST

Application

Securityservice

HOST

Application

Securityservice

2

3

4

1

Figure 14.4 Automatic Key Distribution for Connection-Oriented Protocol


session key distribution. Thus, there may need to be as many as [n(n - 1)]/2 master keys for a configuration with n end systems.

A session key may be established with the following sequence of steps (Figure 14.5).

1. A issues a request to B for a session key and includes a nonce, N1.

2. B responds with a message that is encrypted using the shared master key. The response includes the session key selected by B, an identifier of B, the value f(N1), and another nonce, N2.

3. Using the new session key, A returns f(N2) to B.

Thus, although each node must maintain at most (n - 1) master keys, as many session keys as required may be generated and used. Because the messages trans-ferred using the master key are short, cryptanalysis is difficult. As before, session keys are used for only a limited time to protect them.

Controlling Key Usage

The concept of a key hierarchy and the use of automated key distribution techniques greatly reduce the number of keys that must be manually managed and distributed. It also may be desirable to impose some control on the way in which automatically distributed keys are used. For example, in addition to separating master keys from session keys, we may wish to define different types of session keys on the basis of use, such as

• Data-encrypting key, for general communication across a network

• PIN-encrypting key, for personal identification numbers (PINs) used in elec-tronic funds transfer and point-of-sale applications

• File-encrypting key, for encrypting files stored in publicly accessible locations

To illustrate the value of separating keys by type, consider the risk that a mas-ter key is imported as a data-encrypting key into a device. Normally, the master key is physically secured within the cryptographic hardware of the key distribution cen-ter and of the end systems. Session keys encrypted with this master key are available to application programs, as are the data encrypted with such session keys. However, if a master key is treated as a session key, it may be possible for an unauthorized application to obtain plaintext of session keys encrypted with that master key.

(1) IDA || N1

(2) E(Km, [Ks || IDA || IDB || f(N1) || N2 ])

InitiatorA

ResponderB

(3) E(Ks, f(N2))

Figure 14.5 Decentralized Key Distribution


Thus, it may be desirable to institute controls in systems that limit the ways in which keys are used, based on characteristics associated with those keys. One simple plan is to associate a tag with each key ([JONE82]; see also [DAVI89]). The proposed technique is for use with DES and makes use of the extra 8 bits in each 64-bit DES key. That is, the eight non-key bits ordinarily reserved for parity check-ing form the key tag. The bits have the following interpretation:

• One bit indicates whether the key is a session key or a master key.

• One bit indicates whether the key can be used for encryption.

• One bit indicates whether the key can be used for decryption.

• The remaining bits are spares for future use.

Because the tag is embedded in the key, it is encrypted along with the key when that key is distributed, thus providing protection. The drawbacks of this scheme are

1. The tag length is limited to 8 bits, limiting its flexibility and functionality.

2. Because the tag is not transmitted in clear form, it can be used only at the point of decryption, limiting the ways in which key use can be controlled.

A more flexible scheme, referred to as the control vector, is described in [MATY91a and b]. In this scheme, each session key has an associated control vector consisting of a number of fields that specify the uses and restrictions for that session key. The length of the control vector may vary.

The control vector is cryptographically coupled with the key at the time of key generation at the KDC. The coupling and decoupling processes are illustrated in Figure 14.6. As a first step, the control vector is passed through a hash function that produces a value whose length is equal to the encryption key length. Hash functions are discussed in detail in Chapter 11. In essence, a hash function maps values from a larger range into a smaller range with a reasonably uniform spread. Thus, for example, if numbers in the range 1 to 100 are hashed into numbers in the range 1 to 10, approximately 10% of the source values should map into each of the target values.

The hash value is then XORed with the master key to produce an output that is used as the key input for encrypting the session key. Thus,

Hash value = H = h(CV)

Key input = Km ⊕ H

Ciphertext = E([Km ⊕ H], Ks)

where Km is the master key and Ks is the session key. The session key is recovered in plaintext by the reverse operation:

D([Km ⊕ H], E([Km ⊕ H], Ks))

When a session key is delivered to a user from the KDC, it is accompanied by the control vector in clear form. The session key can be recovered only by using both the master key that the user shares with the KDC and the control vector. Thus, the linkage between the session key and its control vector is maintained.

14.2 / syMMetriC Key distribution using asyMMetriC enCryption 427

Use of the control vector has two advantages over use of an 8-bit tag. First, there is no restriction on length of the control vector, which enables arbitrarily complex con-trols to be imposed on key use. Second, the control vector is available in clear form at all stages of operation. Thus, control of key use can be exercised in multiple locations.

14.2 Symmetric Key DiStribution uSing aSymmetric encryption

Because of the inefficiency of public-key cryptosystems, they are almost never used for the direct encryption of sizable block of data, but are limited to relatively small blocks. One of the most important uses of a public-key cryptosystem is to encrypt secret keys for distribution. We see many specific examples of this in Part Five. Here, we discuss general principles and typical approaches.

Simple Secret Key Distribution

An extremely simple scheme was put forward by Merkle [MERK79], as illustrated in Figure 14.7. If A wishes to communicate with B, the following procedure is employed:

1. A generates a public/private key pair {PUa, PRa} and transmits a message to B consisting of PUa and an identifier of A, IDA.

2. B generates a secret key, Ks, and transmits it to A, which is encrypted with A’s public key.

Controlvector

Masterkey

Sessionkey

HashingFunction

Keyinput

Keyinput

Ciphertextinput

Plaintextinput

EncryptionFunction

Encryptedsession key

(a) Control vector encryption

Controlvector

Masterkey

Encryptedsession key

HashingFunction

DecryptionFunction

Session key

(b) Control vector decryption

Figure 14.6 Control Vector Encryption and Decryption


3. A computes D(PRa, E(PUa, Ks)) to recover the secret key. Because only A can decrypt the message, only A and B will know the identity of Ks.

4. A discards PUa and PRa and B discards PUa.

A and B can now securely communicate using conventional encryption and the session key Ks. At the completion of the exchange, both A and B discard Ks. Despite its simplicity, this is an attractive protocol. No keys exist before the start of the communication and none exist after the completion of communication. Thus, the risk of compromise of the keys is minimal. At the same time, the communication is secure from eavesdropping.

The protocol depicted in Figure 14.7 is insecure against an adversary who can intercept messages and then either relay the intercepted message or substitute another message (see Figure 1.3c). Such an attack is known as a man-in-the-middle attack [RIVE84]. We saw this type of attack in Chapter 10 (Figure 10.2). In the present case, if an adversary, D, has control of the intervening communication chan-nel, then D can compromise the communication in the following fashion without being detected (Figure 14.8).

1. A generates a public/private key pair {PUa, PRa} and transmits a message intended for B consisting of PUa and an identifier of A, IDA.

2. D intercepts the message, creates its own public/private key pair {PUd, PRd} and transmits PUs 0 0 IDA to B.

3. B generates a secret key, Ks, and transmits E(PUs, Ks).

4. D intercepts the message and learns Ks by computing D(PRd, E(PUd, Ks)).

5. D transmits E(PUa, Ks) to A.

The result is that both A and B know Ks and are unaware that Ks has also been revealed to D. A and B can now exchange messages using Ks. D no longer actively interferes with the communications channel but simply eavesdrops. Knowing Ks, S can decrypt all messages, and both A and B are unaware of the problem. Thus, this sim-ple protocol is only useful in an environment where the only threat is eavesdropping.

Secret Key Distribution with Confidentiality and Authentication

Figure 14.9, based on an approach suggested in [NEED78], provides protection against both active and passive attacks. We begin at a point when it is assumed that A and B have exchanged public keys by one of the schemes described subsequently in this chapter. Then the following steps occur.

BA

(1) PUa || IDA

(2) E(PUa, Ks)

Figure 14.7 Simple Use of Public-Key Encryption to Establish a Session Key

14.2 / syMMetriC Key distribution using asyMMetriC enCryption 429

Alice Darth Bob

Private key PRAPublic key PUA

Private key PRBPublic key PUBSecret key Ks

Ks =D(PRD, E(PUD, Ks))

Private key PRDPublic key PUD

PUA, IDA

PUD, IDA

E(PUD, Ks)

Alice, Bob, andDarth share K1

E(PUA, Ks)

Figure 14.8 Another Man-in-the-Middle Attack

InitiatorA

ResponderB

(1) E(PUb, [N1 || IDA])

(4) E(PUb, E(PRa, Ks))

(3) E(PUb, N2)

(2) E(PUa, [N1 || N2])

Figure 14.9 Public-Key Distribution of Secret Keys


1. A uses B’s public key to encrypt a message to B containing an identifier of A(IDA) and a nonce (N1), which is used to identify this transaction uniquely.

2. B sends a message to A encrypted with PUa and containing A’s nonce (N1) as well as a new nonce generated by B (N2). Because only B could have decrypted message (1), the presence of N1 in message (2) assures A that the correspon-dent is B.

3. A returns N2, encrypted using B’s public key, to assure B that its correspon-dent is A.

4. A selects a secret key Ks and sends M = E(PUb, E(PRa, Ks)) to B. Encryption of this message with B’s public key ensures that only B can read it; encryption with A’s private key ensures that only A could have sent it.

5. B computes D(PUa, D(PRb, M)) to recover the secret key.

The result is that this scheme ensures both confidentiality and authentication in the exchange of a secret key.

A Hybrid Scheme

Yet another way to use public-key encryption to distribute secret keys is a hybrid approach in use on IBM mainframes [LE93]. This scheme retains the use of a key distribution center (KDC) that shares a secret master key with each user and dis-tributes secret session keys encrypted with the master key. A public-key scheme is used to distribute the master keys. The following rationale is provided for using this three-level approach:

• Performance: There are many applications, especially transaction-oriented applications, in which the session keys change frequently. Distribution of ses-sion keys by public-key encryption could degrade overall system performance because of the relatively high computational load of public-key encryption and decryption. With a three-level hierarchy, public-key encryption is used only occasionally to update the master key between a user and the KDC.

• Backward compatibility: The hybrid scheme is easily overlaid on an existing KDC scheme with minimal disruption or software changes.

The addition of a public-key layer provides a secure, efficient means of dis-tributing master keys. This is an advantage in a configuration in which a single KDC serves a widely distributed set of users.

14.3 DiStribution of pubLic KeyS

Several techniques have been proposed for the distribution of public keys. Virtually all these proposals can be grouped into the following general schemes:

• Public announcement • Publicly available directory • Public-key authority • Public-key certificates

14.3 / distribution of publiC Keys 431

Public Announcement of Public Keys

On the face of it, the point of public-key encryption is that the public key is public. Thus, if there is some broadly accepted public-key algorithm, such as RSA, any partic-ipant can send his or her public key to any other participant or broadcast the key to the community at large (Figure 14.10). For example, because of the growing popularity of PGP (pretty good privacy, discussed in Chapter 19), which makes use of RSA, many PGP users have adopted the practice of appending their public key to messages that they send to public forums, such as USENET newsgroups and Internet mailing lists.

Although this approach is convenient, it has a major weakness. Anyone can forge such a public announcement. That is, some user could pretend to be user A and send a public key to another participant or broadcast such a public key. Until such time as user A discovers the forgery and alerts other participants, the forger is able to read all encrypted messages intended for A and can use the forged keys for authentication (see Figure 9.3).

Publicly Available Directory

A greater degree of security can be achieved by maintaining a publicly available dynamic directory of public keys. Maintenance and distribution of the public directory would have to be the responsibility of some trusted entity or organization (Figure 14.11). Such a scheme would include the following elements:

1. The authority maintains a directory with a {name, public key} entry for each participant.

2. Each participant registers a public key with the directory authority. Registration would have to be in person or by some form of secure authenti-cated communication.

3. A participant may replace the existing key with a new one at any time, either because of the desire to replace a public key that has already been used for a large amount of data, or because the corresponding private key has been com-promised in some way.

4. Participants could also access the directory electronically. For this purpose, secure, authenticated communication from the authority to the participant is mandatory.

PUa

PUa

PUa

PUa

PUb

PUb

PUb

PUb

BA

Figure 14.10 Uncontrolled Public-Key Distribution


This scheme is clearly more secure than individual public announcements but still has vulnerabilities. If an adversary succeeds in obtaining or computing the private key of the directory authority, the adversary could authoritatively pass out counterfeit public keys and subsequently impersonate any participant and eaves-drop on messages sent to any participant. Another way to achieve the same end is for the adversary to tamper with the records kept by the authority.

Public-Key Authority

Stronger security for public-key distribution can be achieved by providing tighter control over the distribution of public keys from the directory. A typical scenario is illustrated in Figure 14.12, which is based on a figure in [POPE79]. As before, the scenario assumes that a central authority maintains a dynamic directory of public keys of all participants. In addition, each participant reliably knows a public key for the authority, with only the authority knowing the corre-sponding private key. The following steps (matched by number to Figure 14.12) occur.

1. A sends a timestamped message to the public-key authority containing a request for the current public key of B.

2. The authority responds with a message that is encrypted using the author-ity’s private key, PRauth. Thus, A is able to decrypt the message using the authority’s public key. Therefore, A is assured that the message originated with the authority. The message includes the following:

• B’s public key, PUb, which A can use to encrypt messages destined for B

• The original request used to enable A to match this response with the cor-responding earlier request and to verify that the original request was not altered before reception by the authority

• The original timestamp given so A can determine that this is not an old message from the authority containing a key other than B’s current public key

Public-keydirectory

PUa PUb

A B

Figure 14.11 Public-Key Publication

14.3 / distribution of publiC Keys 433

3. A stores B’s public key and also uses it to encrypt a message to B containing an identifier of A (IDA) and a nonce (N1), which is used to identify this trans-action uniquely.

4, 5. B retrieves A’s public key from the authority in the same manner as A retrieved B’s public key.At this point, public keys have been securely delivered to A and B, and they

may begin their protected exchange. However, two additional steps are desirable:

6. B sends a message to A encrypted with PUa and containing A’s nonce (N1) as well as a new nonce generated by B (N2). Because only B could have decrypted message (3), the presence of N1 in message (6) assures A that the correspondent is B.

7. A returns N2, which is encrypted using B’s public key, to assure B that its correspondent is A.

Thus, a total of seven messages are required. However, the initial five mes-sages need be used only infrequently because both A and B can save the other’s public key for future use—a technique known as caching. Periodically, a user should request fresh copies of the public keys of its correspondents to ensure currency.

Public-Key Certificates

The scenario of Figure 14.12 is attractive, yet it has some drawbacks. The public-key authority could be somewhat of a bottleneck in the system, for a user must appeal to the authority for a public key for every other user that it wishes to contact. As before, the directory of names and public keys maintained by the authority is vul-nerable to tampering.

Public-keyAuthorityInitiator A Responder B

(1) Request || T1

(2) E(PRauth, [PUb || Request || T1])

(3) E(PUb, [ IDA || N1])

(4) Request || T2

(5) E(PRauth, [PUa || Request || T2])

(6) E(PUa, [ N1 || N2])

(7) E(PUb, N2)

Figure 14.12 Public-Key Distribution Scenario


An alternative approach, first suggested by Kohnfelder [KOHN78], is to use certificates that can be used by participants to exchange keys without contacting a public-key authority, in a way that is as reliable as if the keys were obtained directly from a public-key authority. In essence, a certificate consists of a public key, an identifier of the key owner, and the whole block signed by a trusted third party. Typically, the third party is a certificate authority, such as a government agency or a financial institution, that is trusted by the user community. A user can present his or her public key to the authority in a secure manner and obtain a certificate. The user can then publish the certificate. Anyone needing this user’s public key can obtain the certificate and verify that it is valid by way of the attached trusted signature. A participant can also convey its key information to another by transmitting its certificate. Other participants can verify that the certificate was created by the authority. We can place the following requirements on this scheme:

1. Any participant can read a certificate to determine the name and public key of the certificate’s owner.

2. Any participant can verify that the certificate originated from the certificate authority and is not counterfeit.

3. Only the certificate authority can create and update certificates.

These requirements are satisfied by the original proposal in [KOHN78]. Denning [DENN83] added the following additional requirement:

4. Any participant can verify the currency of the certificate.

A certificate scheme is illustrated in Figure 14.13. Each participant applies to the certificate authority, supplying a public key and requesting a certificate. Application must be in person or by some form of secure authenticated communi-cation. For participant A, the authority provides a certificate of the form

CA = E(PRauth, [T } IDA }PUa])

where PRauth is the private key used by the authority and T is a timestamp. A may then pass this certificate on to any other participant, who reads and verifies the cer-tificate as follows:

D(PUauth, CA) = D(PUauth, E(PRauth, [T } IDA }PUa])) = (T } IDA }PUa)

The recipient uses the authority’s public key, PUauth, to decrypt the certificate. Because the certificate is readable only using the authority’s public key, this verifies that the certificate came from the certificate authority. The elements IDA and PUa provide the recipient with the name and public key of the certificate’s holder. The timestamp T validates the currency of the certificate. The timestamp counters the following scenario. A’s private key is learned by an adversary. A generates a new private/public key pair and applies to the certificate authority for a new certificate. Meanwhile, the adversary replays the old certificate to B. If B then encrypts mes-sages using the compromised old public key, the adversary can read those messages.

In this context, the compromise of a private key is comparable to the loss of a credit card. The owner cancels the credit card number but is at risk until all possible communicants are aware that the old credit card is obsolete. Thus, the timestamp

14.4 / X.509 CertifiCates 435

serves as something like an expiration date. If a certificate is sufficiently old, it is assumed to be expired.

One scheme has become universally accepted for formatting public-key cer-tificates: the X.509 standard. X.509 certificates are used in most network security applications, including IP security, transport layer security (TLS), and S/MIME, all of which are discussed in Part Five. X.509 is examined in detail in the next section.

14.4 X.509 certificateS

ITU-T recommendation X.509 is part of the X.500 series of recommendations that define a directory service. The directory is, in effect, a server or distributed set of servers that maintains a database of information about users. The information includes a mapping from user name to network address, as well as other attributes and information about the users.

X.509 defines a framework for the provision of authentication services by the X.500 directory to its users. The directory may serve as a repository of public-key certificates of the type discussed in Section 14.3. Each certificate contains the public key of a user and is signed with the private key of a trusted certification authority. In addition, X.509 defines alternative authentication protocols based on the use of public-key certificates.

(a) Obtaining certificates from CA

(b) Exchanging certificates

PUa PUb

A B

CertificateAuthority

CA = E(PRauth, [T1 || IDA || PUa])

CB = E(PRauth, [T2 || IDB || PUb])

(1) CA

(2) CB

A B

Figure 14.13 Exchange of Public-Key Certificates


X.509 is an important standard because the certificate structure and authenti-cation protocols defined in X.509 are used in a variety of contexts. For example, the X.509 certificate format is used in S/MIME (Chapter 19), IP Security (Chapter 20), and SSL/TLS (Chapter 17).

X.509 was initially issued in 1988. The standard was subsequently revised to address some of the security concerns documented in [IANS90] and [MITC90]; a revised recommendation was issued in 1993. A third version was issued in 1995 and revised in 2000.

X.509 is based on the use of public-key cryptography and digital signatures. The standard does not dictate the use of a specific algorithm but recommends RSA. The digital signature scheme is assumed to require the use of a hash func-tion. Again, the standard does not dictate a specific hash algorithm. The 1988 recommendation included the description of a recommended hash algorithm; this algorithm has since been shown to be insecure and was dropped from the 1993 recommendation. Figure 14.14 illustrates the generation of a public-key certificate.

Certificates

The heart of the X.509 scheme is the public-key certificate associated with each user. These user certificates are assumed to be created by some trusted certification authority (CA) and placed in the directory by the CA or by the user. The directory server itself is not responsible for the creation of public keys or for the certifica-tion function; it merely provides an easily accessible location for users to obtain certificates.

Figure 14.15a shows the general format of a certificate, which includes the fol-lowing elements.

Unsigned certificate:contains user ID,user's public key

Signed certificate

Recipient can verifysignature by comparinghash code values

Generate hashcode of unsignedcertificate

Encrypt hash codewith CA's private keyto form signature

H

H

Bob's IDinformation

CAinformation

Bob's public key

E D

Decrypt signaturewith CA's public keyto recover hash code

Use certificate toverify Bob's public key

Create signeddigital certificate

Figure 14.14 Public-Key Certificate Use


• Version: Differentiates among successive versions of the certificate format; the default is version 1. If the issuer unique identifier or subject unique identi-fier are present, the value must be version 2. If one or more extensions are present, the version must be version 3.

• Serial number: An integer value unique within the issuing CA that is unam-biguously associated with this certificate.

• Signature algorithm identifier: The algorithm used to sign the certificate together with any associated parameters. Because this information is repeated in the signature field at the end of the certificate, this field has little, if any, utility.

• Issuer name: X.500 name of the CA that created and signed this certificate.

• Period of validity: Consists of two dates: the first and last on which the certifi-cate is valid.

• Subject name: The name of the user to whom this certificate refers. That is, this certificate certifies the public key of the subject who holds the correspond-ing private key.

• Subject’s public-key information: The public key of the subject, plus an identi-fier of the algorithm for which this key is to be used, together with any associ-ated parameters.

• Issuer unique identifier: An optional-bit string field used to identify uniquely the issuing CA in the event the X.500 name has been reused for different entities.

Certificateserial number

Version

Issuer name

Signaturealgorithmidentifier

Subject name

Extensions

Issuer uniqueidentifier

Subject uniqueidentifier

Algorithm

Parameters

Not before

AlgorithmsParameters

Key

AlgorithmsParameters

Encrypted hash

(a) X.509 certificate

Not after

Subject'spublic key

info

Signature

Period ofvalidity

Ver

sion

1

Ver

sion

2

Ver

sion

3

All

vers

ions

Issuer Name

This update date

Next update date

•••

Signaturealgorithmidentifier

Algorithm

Parameters

User certificate serial #

(b) Certificate revocation list

Revocation date

AlgorithmsParametersEncrypted

Signature

Revokedcertificate

User certificate serial #

Revocation dateRevoked

certificate

Figure 14.15 X.509 Formats


• Subject unique identifier: An optional-bit string field used to identify uniquely the subject in the event the X.500 name has been reused for different entities.

• Extensions: A set of one or more extension fields. Extensions were added in version 3 and are discussed later in this section.

• Signature: Covers all of the other fields of the certificate; it contains the hash code of the other fields encrypted with the CA’s private key. This field includes the signature algorithm identifier.

The unique identifier fields were added in version 2 to handle the possible reuse of subject and/or issuer names over time. These fields are rarely used.

The standard uses the following notation to define a certificate:

CA V A W = CA {V, SN, AI, CA, UCA, A, UA, Ap, TA}

where

Y VXW = the certificate of user X issued by certification authority Y

Y {I} = the signing of I by Y. It consists of I with an encrypted hash code appended

V = version of the certificate

SN = serial number of the certificate

AI = identifier of the algorithm used to sign the certificate

CA = name of certificate authority

UCA = optional unique identifier of the CA

A = name of user A

UA = optional unique identifier of the user A

Ap = public key of user A

TA = period of validity of the certificate

The CA signs the certificate with its private key. If the corresponding public key is known to a user, then that user can verify that a certificate signed by the CA is valid. This is the typical digital signature approach illustrated in Figure 13.2.

Obtaining a User’s CertifiCate User certificates generated by a CA have the fol-lowing characteristics:

• Any user with access to the public key of the CA can verify the user public key that was certified.

• No party other than the certification authority can modify the certificate with-out this being detected.

Because certificates are unforgeable, they can be placed in a directory without the need for the directory to make special efforts to protect them.

If all users subscribe to the same CA, then there is a common trust of that CA. All user certificates can be placed in the directory for access by all users. In addi-tion, a user can transmit his or her certificate directly to other users. In either case, once B is in possession of A’s certificate, B has confidence that messages it encrypts


with A’s public key will be secure from eavesdropping and that messages signed with A’s private key are unforgeable.

If there is a large community of users, it may not be practical for all users to subscribe to the same CA. Because it is the CA that signs certificates, each partici-pating user must have a copy of the CA’s own public key to verify signatures. This public key must be provided to each user in an absolutely secure (with respect to integrity and authenticity) way so that the user has confidence in the associated cer-tificates. Thus, with many users, it may be more practical for there to be a number of CAs, each of which securely provides its public key to some fraction of the users.

Now suppose that A has obtained a certificate from certification authority X1 and B has obtained a certificate from CA X2. If A does not securely know the public key of X2, then B’s certificate, issued by X2, is useless to A. A can read B’s cer-tificate, but A cannot verify the signature. However, if the two CAs have securely exchanged their own public keys, the following procedure will enable A to obtain B’s public key.

Step 1 A obtains from the directory the certificate of X2 signed by X1. Because A securely knows X1’s public key, A can obtain X2’s public key from its certifi-cate and verify it by means of X1’s signature on the certificate.

Step 2 A then goes back to the directory and obtains the certificate of B signed by X2. Because A now has a trusted copy of X2’s public key, A can verify the signature and securely obtain B’s public key.

A has used a chain of certificates to obtain B’s public key. In the notation of X.509, this chain is expressed as

X1 V X2 W X2 V B WIn the same fashion, B can obtain A’s public key with the reverse chain:

X2 V X1 W X1 V A W

This scheme need not be limited to a chain of two certificates. An arbitrarily long path of CAs can be followed to produce a chain. A chain with N elements would be expressed as

X1 V X2 W X2 V X3 W cXN V B W

In this case, each pair of CAs in the chain (Xi, Xi+1) must have created certifi-cates for each other.

All these certificates of CAs by CAs need to appear in the directory, and the user needs to know how they are linked to follow a path to another user’s public-key certificate. X.509 suggests that CAs be arranged in a hierarchy so that naviga-tion is straightforward.

Figure 14.16, taken from X.509, is an example of such a hierarchy. The con-nected circles indicate the hierarchical relationship among the CAs; the associated boxes indicate certificates maintained in the directory for each CA entry. The direc-tory entry for each CA includes two types of certificates:

• Forward certificates: Certificates of X generated by other CAs

• Reverse certificates: Certificates generated by X that are the certificates of other CAs


In this example, user A can acquire the following certificates from the direc-tory to establish a certification path to B:

X V W W W V V W V V Y W Y V Z W Z V B W

When A has obtained these certificates, it can unwrap the certification path in sequence to recover a trusted copy of B’s public key. Using this public key, A can send encrypted messages to B. If A wishes to receive encrypted messages back from B, or to sign messages sent to B, then B will require A’s public key, which can be obtained from the following certification path:

Z V Y W Y V V W V V W W W V X W X V A W

B can obtain this set of certificates from the directory, or A can provide them as part of its initial message to B.

revOCatiOn Of CertifiCates Recall from Figure 14.15 that each certificate includes a period of validity, much like a credit card. Typically, a new certificate is issued just before the expiration of the old one. In addition, it may be desir-able on occasion to revoke a certificate before it expires, for one of the following reasons.

U

V

W Y

Z

B

X

C A

U<<V>>V<<U>>

V<<W>>W<<V>>

V<<Y>>Y<<V>>

W<<X>>X<<W>>X<<Z>>

Y<<Z>>Z<<Y>>Z<<X>>

X<<C>> X<<A>> Z<<B>>

Figure 14.16 X.509 Hierarchy: A Hypothetical Example


1. The user’s private key is assumed to be compromised. 2. The user is no longer certified by this CA. Reasons for this include that the

subject’s name has changed, the certificate is superseded, or the certificate was not issued in conformance with the CA’s policies.

3. The CA’s certificate is assumed to be compromised.

Each CA must maintain a list consisting of all revoked but not expired cer-tificates issued by that CA, including both those issued to users and to other CAs. These lists should also be posted on the directory.

Each certificate revocation list (CRL) posted to the directory is signed by the issuer and includes (Figure 14.15b) the issuer’s name, the date the list was created, the date the next CRL is scheduled to be issued, and an entry for each revoked certificate. Each entry consists of the serial number of a certificate and revocation date for that certificate. Because serial numbers are unique within a CA, the serial number is sufficient to identify the certificate.

When a user receives a certificate in a message, the user must determine whether the certificate has been revoked. The user could check the directory each time a certificate is received. To avoid the delays (and possible costs) associated with directory searches, it is likely that the user would maintain a local cache of cer-tificates and lists of revoked certificates.

X.509 Version 3

The X.509 version 2 format does not convey all of the information that recent design and implementation experience has shown to be needed. [FORD95] lists the follow-ing requirements not satisfied by version 2.

1. The subject field is inadequate to convey the identity of a key owner to a pub-lic-key user. X.509 names may be relatively short and lacking in obvious iden-tification details that may be needed by the user.

2. The subject field is also inadequate for many applications, which typically rec-ognize entities by an Internet e-mail address, a URL, or some other Internet-related identification.

3. There is a need to indicate security policy information. This enables a security ap-plication or function, such as IPSec, to relate an X.509 certificate to a given policy.

4. There is a need to limit the damage that can result from a faulty or malicious CA by setting constraints on the applicability of a particular certificate.

5. It is important to be able to identify different keys used by the same owner at different times. This feature supports key lifecycle management: in particular, the ability to update key pairs for users and CAs on a regular basis or under exceptional circumstances.

Rather than continue to add fields to a fixed format, standards developers felt that a more flexible approach was needed. Thus, version 3 includes a number of optional extensions that may be added to the version 2 format. Each extension consists of an extension identifier, a criticality indicator, and an extension value. The criticality indicator indicates whether an extension can be safely ignored. If the indicator has a value of TRUE and an implementation does not recognize the extension, it must treat the certificate as invalid.


The certificate extensions fall into three main categories: key and policy infor-mation, subject and issuer attributes, and certification path constraints.

Key and POliCy infOrmatiOn These extensions convey additional information about the subject and issuer keys, plus indicators of certificate policy. A certificate policy is a named set of rules that indicates the applicability of a certificate to a par-ticular community and/or class of application with common security requirements. For example, a policy might be applicable to the authentication of electronic data interchange (EDI) transactions for the trading of goods within a given price range.

This area includes:

• Authority key identifier: Identifies the public key to be used to verify the sig-nature on this certificate or CRL. Enables distinct keys of the same CA to be differentiated. One use of this field is to handle CA key pair updating.

• Subject key identifier: Identifies the public key being certified. Useful for sub-ject key pair updating. Also, a subject may have multiple key pairs and, corre-spondingly, different certificates for different purposes (e.g., digital signature and encryption key agreement).

• Key usage: Indicates a restriction imposed as to the purposes for which, and the policies under which, the certified public key may be used. May indicate one or more of the following: digital signature, nonrepudiation, key encryp-tion, data encryption, key agreement, CA signature verification on certifi-cates, CA signature verification on CRLs.

• Private-key usage period: Indicates the period of use of the private key cor-responding to the public key. Typically, the private key is used over a different period from the validity of the public key. For example, with digital signature keys, the usage period for the signing private key is typically shorter than that for the verifying public key.

• Certificate policies: Certificates may be used in environments where multiple policies apply. This extension lists policies that the certificate is recognized as supporting, together with optional qualifier information.

• Policy mappings: Used only in certificates for CAs issued by other CAs. Policy mappings allow an issuing CA to indicate that one or more of that issuer’s policies can be considered equivalent to another policy used in the subject CA’s domain.

CertifiCate sUbjeCt and issUer attribUtes These extensions support alternative names, in alternative formats, for a certificate subject or certificate issuer and can convey additional information about the certificate subject to increase a certificate user’s confidence that the certificate subject is a particular person or entity. For example, information such as postal address, position within a corporation, or pic-ture image may be required.

The extension fields in this area include:

• Subject alternative name: Contains one or more alternative names, using any of a variety of forms. This field is important for supporting certain applications, such as electronic mail, EDI, and IPSec, which may employ their own name forms.

14.5 / publiC-Key infrastruCture 443

• Issuer alternative name: Contains one or more alternative names, using any of a variety of forms.

• Subject directory attributes: Conveys any desired X.500 directory attribute values for the subject of this certificate.

CertifiCatiOn Path COnstraints These extensions allow constraint specifications to be included in certificates issued for CAs by other CAs. The constraints may restrict the types of certificates that can be issued by the subject CA or that may occur subsequently in a certification chain.

The extension fields in this area include:

• Basic constraints: Indicates if the subject may act as a CA. If so, a certification path length constraint may be specified.

• Name constraints: Indicates a name space within which all subject names in subsequent certificates in a certification path must be located.

• Policy constraints: Specifies constraints that may require explicit certificate policy identification or inhibit policy mapping for the remainder of the certifi-cation path.

14.5 pubLic-Key infraStructure

RFC 4949 (Internet Security Glossary) defines public-key infrastructure (PKI) as the set of hardware, software, people, policies, and procedures needed to create, manage, store, distribute, and revoke digital certificates based on asymmetric cryp-tography. The principal objective for developing a PKI is to enable secure, conve-nient, and efficient acquisition of public keys. The Internet Engineering Task Force (IETF) Public Key Infrastructure X.509 (PKIX) working group has been the driv-ing force behind setting up a formal (and generic) model based on X.509 that is suitable for deploying a certificate-based architecture on the Internet. This section describes the PKIX model.

Figure 14.17 shows the interrelationship among the key elements of the PKIX model. These elements are

• End entity: A generic term used to denote end users, devices (e.g., servers, routers), or any other entity that can be identified in the subject field of a public-key certificate. End entities typically consume and/or support PKI-related services.

• Certification authority (CA): The issuer of certificates and (usually) certifi-cate revocation lists (CRLs). It may also support a variety of administrative functions, although these are often delegated to one or more Registration Authorities.

• Registration authority (RA): An optional component that can assume a number of administrative functions from the CA. The RA is often associated with the end entity registration process but can assist in a number of other areas as well.


• CRL issuer: An optional component that a CA can delegate to publish CRLs.

• Repository: A generic term used to denote any method for storing certificates and CRLs so that they can be retrieved by end entities.

PKIX Management Functions

PKIX identifies a number of management functions that potentially need to be sup-ported by management protocols. These are indicated in Figure 14.17 and include the following:

• Registration: This is the process whereby a user first makes itself known to a CA (directly or through an RA), prior to that CA issuing a certificate or certificates for that user. Registration begins the process of enrolling in a PKI. Registration usually involves some offline or online procedure for mutual authentication. Typically, the end entity is issued one or more shared secret keys used for subsequent authentication.

• Initialization: Before a client system can operate securely, it is necessary to install key materials that have the appropriate relationship with keys stored elsewhere in the infrastructure. For example, the client needs to be securely initialized with the public key and other assured information of the trusted CA(s), to be used in validating certificate paths.

End entityCertificate/CRL retrieval

Certificatepublication

Certificate/CRLpublication

CRLpublication

Crosscertification

Cer

tifi

cate

/CR

L R

epos

itor

y

Certificateauthority

Registrationauthority

Certificateauthority

Registration,initialization,certification,key pair recovery,key pair updaterevocation request

PKIusers

PKImanagement

entities

CRL issuer

Figure 14.17 PKIX Architectural Model

14.6 / reCoMMended reading 445

• Certification: This is the process in which a CA issues a certificate for a user’s public key, returns that certificate to the user’s client system, and/or posts that certificate in a repository.

• Key pair recovery: Key pairs can be used to support digital signature creation and verification, encryption and decryption, or both. When a key pair is used for encryption/decryption, it is important to provide a mechanism to recover the necessary decryption keys when normal access to the keying material is no longer possible, otherwise it will not be possible to recover the encrypted data. Loss of access to the decryption key can result from forgotten passwords/PINs, corrupted disk drives, damage to hardware tokens, and so on. Key pair recovery allows end entities to restore their encryption/decryption key pair from an authorized key backup facility (typically, the CA that issued the end entity’s certificate).

• Key pair update: All key pairs need to be updated regularly (i.e., replaced with a new key pair) and new certificates issued. Update is required when the certificate lifetime expires and as a result of certificate revocation.

• Revocation request: An authorized person advises a CA of an abnormal situ-ation requiring certificate revocation. Reasons for revocation include private-key compromise, change in affiliation, and name change.

• Cross certification: Two CAs exchange information used in establishing a cross-certificate. A cross-certificate is a certificate issued by one CA to another CA that contains a CA signature key used for issuing certificates.

PKIX Management Protocols

The PKIX working group has defines two alternative management protocols between PKIX entities that support the management functions listed in the pre-ceding subsection. RFC 2510 defines the certificate management protocols (CMP). Within CMP, each of the management functions is explicitly identified by specific protocol exchanges. CMP is designed to be a flexible protocol able to accommodate a variety of technical, operational, and business models.

RFC 2797 defines certificate management messages over CMS (CMC), where CMS refers to RFC 2630, cryptographic message syntax. CMC is built on earlier work and is intended to leverage existing implementations. Although all of the PKIX functions are supported, the functions do not all map into specific protocol exchanges.

14.6 recommenDeD reaDing

An exhaustive and essential resource on the topics of this chapter is the three-volume NIST SP800-57 [BARK12. BARK05, BARK09]. [FUMY93] is a good survey of key management principles. Another interesting survey, which looks at many key management techniques, is [HEGL06].

[PERL99] reviews various trust models that can be used in a PKI. [GUTM02] high-lights difficulties in PKI use and recommends approaches for an effective PKI.


14.7 Key termS, review QueStionS, anD probLemS

Key Terms

end-to-end encryptionkey distributionkey distribution center (KDC)key management

man-in-the-middle attackmaster keynoncepublic-key certificate

public-key directoryX.509 certificate

BARK12 Barker, E., et al. Recommendation for Key Management—Part 1: General. NIST SP800-57, June 2012.

BARK05 Barker, E., et al. Recommendation for Key Management—Part 2: Best Practices for Key Management Organization. NIST SP800-57, August 2005.

BARK09 Barker, E., et al. Recommendation for Key Management—Part 3: Specific Key Management Guidance. NIST SP800-57, December 2009.

FUMY93 Fumy, S., and Landrock, P. “Principles of Key Management.” IEEE Journal on Selected Areas in Communications, June 1993.

GUTM02 Gutmann, P. “PKI: It’s Not Dead, Just Resting.” Computer, August 2002.HEGL06 Hegland, A., et al. “A Survey of Key Management in Ad Hoc Networks.”

IEEE Communications Surveys & Tutorials. 3rd Quarter 2006.PERL99 Perlman, R. “An Overview of PKI Trust Models.” IEEE Network, November/

December 1999.

Review Questions 14.1 List ways in which secret keys can be distributed to two communicating parties. 14.2 What is the difference between a session key and a master key? 14.3 What is a nonce? 14.4 What is a key distribution center? 14.5 What are two different uses of public-key cryptography related to key distribution? 14.6 List four general categories of schemes for the distribution of public keys. 14.7 What are the essential ingredients of a public-key directory? 14.8 What is a public-key certificate? 14.9 What are the requirements for the use of a public-key certificate scheme? 14.10 What is the purpose of the X.509 standard? 14.11 What is a chain of certificates? 14.12 How is an X.509 certificate revoked?

Problems 14.1 One local area network vendor provides a key distribution facility, as illustrated in

Figure 14.18. a. Describe the scheme. b. Compare this scheme to that of Figure 14.3. What are the pros and cons?

14.7 / Key terMs, review Questions, and probleMs 447

14.2 “We are under great pressure, Holmes.” Detective Lestrade looked nervous. “We have learned that copies of sensitive government documents are stored in computers of one foreign embassy here in London. Normally these documents exist in electronic form only on a selected few government computers that satisfy the most stringent security re-quirements. However, sometimes they must be sent through the network connecting all government computers. But all messages in this network are encrypted using a top-secret encryption algorithm certified by our best crypto experts. Even the NSA and the KGB are unable to break it. And now these documents have appeared in hands of diplomats of a small, otherwise insignificant, country. And we have no idea how it could happen.”

“But you do have some suspicion who did it, do you?” asked Holmes.“Yes, we did some routine investigation. There is a man who has legal access

to one of the government computers and has frequent contacts with diplomats from the embassy. But the computer he has access to is not one of the trusted ones where these documents are normally stored. He is the suspect, but we have no idea how he could obtain copies of the documents. Even if he could obtain a copy of an encrypted document, he couldn’t decrypt it.”

“Hmm, please describe the communication protocol used on the network.” Holmes opened his eyes, thus proving that he had followed Lestrade’s talk with an attention that contrasted with his sleepy look.

“Well, the protocol is as follows. Each node N of the network has been assigned a unique secret key Kn. This key is used to secure communication between the node and a trusted server. That is, all the keys are stored also on the server. User A, wish-ing to send a secret message M to user B, initiates the following protocol:

1. A generates a random number R and sends to the server his name A, destination B, and E(Ka, R).

2. Server responds by sending E(Kb, R) to A. 3. A sends E(R, M) together with E(Kb, R) to B. 4. B knows Kb, thus decrypts E(Kb, R), to get R and will subsequently use R to

decrypt E(R, M) to get M.

You see that a random key is generated every time a message has to be sent. I admit the man could intercept messages sent between the top-secret trusted nodes, but I see no way he could decrypt them.”

KeyDistribution

Center (KDC)

B A

(1) IDA, E(Ka, Na)

(2) IDA, E(Ka, Na), IDB, E(Kb, Nb)

(4) E(Ka, [Ks, IDB, Na])

(3) E(Kb, [Ks, IDA, Nb]), E(Ka, [Ks, IDB, Na])

Figure 14.18 Figure for Problem 14.1


“Well, I think you have your man, Lestrade. The protocol isn’t secure because the server doesn’t authenticate users who send him a request. Apparently designers of the protocol have believed that sending E(Kx, R) implicitly authenticates user X as the sender, as only X (and the server) knows Kx. But you know that E(Kx, R) can be intercepted and later replayed. Once you understand where the hole is, you will be able to obtain enough evidence by monitoring the man’s use of the computer he has access to. Most likely he works as follows. After intercepting E(Ka, R) and E(R, M) (see steps 1 and 3 of the protocol), the man, let’s denote him as Z, will continue by pretending to be A and …

Finish the sentence for Holmes. 14.3 The 1988 version of X.509 lists properties that RSA keys must satisfy to be secure

given current knowledge about the difficulty of factoring large numbers. The discus-sion concludes with a constraint on the public exponent and the modulus n:

It must be ensured that e 7 log2(n) to prevent attack by taking the eth root mod n to disclose the plaintext.

Although the constraint is correct, the reason given for requiring it is incorrect. What is wrong with the reason given and what is the correct reason?

14.4 Find at least one intermediate certification authority’s certificate and one trusted root certification authority’s certificate on your computer (e.g. in the browser). Print screenshots of both the general and details tab for each certificate.

14.5 NIST defines the term cryptoperiod as the time span during which a specific key is authorized for use or in which the keys for a given system or application may remain in effect. One document on key management uses the following time diagram for a shared secret key.

Originator Usage Period

Recipient Usage Period

Cryptoperiod

Explain the overlap by giving an example application in which the originator’s usage period for the shared secret key begins before the recipient’s usage period and also ends before the recipients usage period.

14.6 Consider the following protocol, designed to let A and B decide on a fresh, shared session key K=

AB. We assume that they already share a long-term key KAB. 1. A S B:A, NA. 2. B S A:E(KAB, [NA, K=

AB]) 3. A S B:E(K=

AB, NA) a. We first try to understand the protocol designer’s reasoning:

— Why would A and B believe after the protocol ran that they share K=AB with the

other party?—Why would they believe that this shared key is fresh?

In both cases, you should explain both the reasons of both A and B, so your answer should complete the sentencesA believes that she shares K=

AB with B since…B believes that he shares K=

AB with A since…A believes that K=

AB is fresh since…B believes that K=

AB is fresh since…

14.7 / Key terMs, review Questions, and probleMs 449

b. Assume now that A starts a run of this protocol with B. However, the connection is intercepted by the adversary C. Show how C can start a new run of the protocol using reflection, causing A to believe that she has agreed on a fresh key with B (in spite of the fact that she has only been communicating with C). Thus, in particular, the belief in (a) is false.

c. Propose a modification of the protocol that prevents this attack. 14.7 What are the core components of a PKI? Briefly describe each component. 14.8 Explain the problems with key management and how it affects symmetric

cryptography.

Note: The remaining problems deal with the a cryptographic product developed by IBM, which is briefly described in a document at this book’s Premium Content Web site (IBMCrypto.pdf). Try these problems after reviewing the document.

14.9 What is the effect of adding the instruction EMKi

EMKi: X S E(KMHi, X) i = 0, 1

14.10 Suppose N different systems use the IBM Cryptographic Subsystem with host master keys KMH[i](i = 1, 2, cN). Devise a method for communicating between systems without requiring the system to either share a common host master key or to divulge their individual host master keys. Hint: each system needs three variants of its host master key.

14.11 The principal objective of the IBM Cryptographic Subsystem is to protect transmis-sions between a terminal and the processing system. Devise a procedure, perhaps adding instructions, which will allow the processor to generate a session key KS and distribute it to Terminal i and Terminal j without having to store a key-equivalent variable in the host.

450450

Chapter

User AUthenticAtion15.1 Remote User-Authentication Principles

Mutual Authentication One-Way Authentication

15.2 Remote User-Authentication Using Symmetric Encryption


15.3 Kerberos

Motivation Kerberos Version 4 Kerberos Version 5

15.4 Remote User Authentication Using Asymmetric Encryption


15.5 Federated Identity Management

Identity Management Identity Federation

15.6 Personal Identity Verification

PIV System Model PIV Documentation PIV Credentials and Keys Authentication



15.1 / Remote useR-AuthenticAtion PRinciPles 451

“Badges? We ain’t got no badges! We don’t need no badges! I don’t have to show you any stinking badges!”

—The Treasure of the Sierra Madre, 1948

Learning Objectives


u Understand the distinction between identification and verification.

u Present an overview of techniques for remote user authentication using symmetric encryption.

u Give a presentation on Kerberos.

u Explain the differences between versions 4 and 5 of Kerberos.

u Describe the use of Kerberos in multiple realms.

u Present an overview of techniques for remote user authentication using asymmetric encryption.

u Understand the need for a federated identity management system.

u Explain the use of PIV mechanisms as part of a user authentication system.

This chapter examines some of the authentication functions that have been developed to support network-based use authentication. The chapter begins with an introduction to some of the concepts and key considerations for user authentication over a network or the Internet. The next section examines user-authentication protocols that rely on symmetric encryption. This is followed by a section on one of the earliest and also one of the most widely used authentication services: Kerberos. Next, the chapter looks at user-authentication protocols that rely on asymmetric encryption. This is followed by a discussion of the X.509 user-authentication protocol. Finally, the concept of federated identity is introduced.

15.1 remOte user-authenticatiOn PrinciPLes

In most computer security contexts, user authentication is the fundamental building block and the primary line of defense. User authentication is the basis for most types of access control and for user accountability. RFC 4949 (Internet Security Glossary) defines user authentication as shown on the following page.

For example, user Alice Toklas could have the user identifier ABTOKLAS. This information needs to be stored on any server or computer system that Alice wishes to use and could be known to system administrators and other users. A typical item of authentication information associated with this user

452 chAPteR 15 / useR AuthenticAtion

ID is a password, which is kept secret (known only to Alice and to the system). If no one is able to obtain or guess Alice’s password, then the combination of Alice’s user ID and password enables administrators to set up Alice’s access per-missions and audit her activity. Because Alice’s ID is not secret, system users can send her e-mail, but because her password is secret, no one can pretend to be Alice.

In essence, identification is the means by which a user provides a claimed identity to the system; user authentication is the means of establishing the validity of the claim. Note that user authentication is distinct from message authentication. As defined in Chapter 12, message authentication is a procedure that allows com-municating parties to verify that the contents of a received message have not been altered and that the source is authentic. This chapter is concerned solely with user authentication.

There are four general means of authenticating a user’s identity, which can be used alone or in combination:

• Something the individual knows: Examples include a password, a personal identification number (PIN), or answers to a prearranged set of questions.

• Something the individual possesses: Examples include cryptographic keys, electronic keycards, smart cards, and physical keys. This type of authenticator is referred to as a token.

• Something the individual is (static biometrics): Examples include recognition by fingerprint, retina, and face.

• Something the individual does (dynamic biometrics): Examples include recog-nition by voice pattern, handwriting characteristics, and typing rhythm.

All of these methods, properly implemented and used, can provide secure user authentication. However, each method has problems. An adversary may be able to guess or steal a password. Similarly, an adversary may be able to forge or steal a token. A user may forget a password or lose a token. Furthermore, there is a significant administrative overhead for managing password and token information on systems and securing such information on systems. With respect to biometric au-thenticators, there are a variety of problems, including dealing with false positives and false negatives, user acceptance, cost, and convenience. For network-based user authentication, the most important methods involve cryptographic keys and some-thing the individual knows, such as a password.

The process of verifying an identity claimed by or for a system entity. An authen-tication process consists of two steps:

• Identification step: Presenting an identifier to the security system. (Identifiers should be assigned carefully, because authenticated identities are the basis for other security services, such as access control service.)

• Verification step: Presenting or generating authentication information that corroborates the binding between the entity and the identifier.

15.1 / Remote useR-AuthenticAtion PRinciPles 453

Mutual Authentication

An important application area is that of mutual authentication protocols. Such protocols enable communicating parties to satisfy themselves mutually about each other’s identity and to exchange session keys. This topic was examined in Chapter 14. There, the focus was key distribution. We return to this topic here to consider the wider implications of authentication.

Central to the problem of authenticated key exchange are two issues: confiden-tiality and timeliness. To prevent masquerade and to prevent compromise of session keys, essential identification and session-key information must be communicated in en-crypted form. This requires the prior existence of secret or public keys that can be used for this purpose. The second issue, timeliness, is important because of the threat of mes-sage replays. Such replays, at worst, could allow an opponent to compromise a session key or successfully impersonate another party. At minimum, a successful replay can disrupt operations by presenting parties with messages that appear genuine but are not.

[GONG93] lists the following examples of replay attacks:

1. The simplest replay attack is one in which the opponent simply copies a message and replays it later.

2. An opponent can replay a timestamped message within the valid time window. If both the original and the replay arrive within then time window, this incident can be logged.

3. As with example (2), an opponent can replay a timestamped message within the valid time window, but in addition, the opponent suppresses the original message. Thus, the repetition cannot be detected.

4. Another attack involves a backward replay without modification. This is a replay back to the message sender. This attack is possible if symmetric encryp-tion is used and the sender cannot easily recognize the difference between messages sent and messages received on the basis of content.

One approach to coping with replay attacks is to attach a sequence number to each message used in an authentication exchange. A new message is accepted only if its sequence number is in the proper order. The difficulty with this approach is that it requires each party to keep track of the last sequence number for each claimant it has dealt with. Because of this overhead, sequence numbers are generally not used for authentication and key exchange. Instead, one of the following two general ap-proaches is used:

• Timestamps: Party A accepts a message as fresh only if the message contains a timestamp that, in A’s judgment, is close enough to A’s knowledge of current time. This approach requires that clocks among the various participants be synchronized.

• Challenge/response: Party A, expecting a fresh message from B, first sends B a nonce (challenge) and requires that the subsequent message (response) received from B contain the correct nonce value.

It can be argued (e.g., [LAM92a]) that the timestamp approach should not be used for connection-oriented applications because of the inherent difficulties with this


technique. First, some sort of protocol is needed to maintain synchronization among the various processor clocks. This protocol must be both fault tolerant, to cope with network errors, and secure, to cope with hostile attacks. Second, the opportunity for a successful attack will arise if there is a temporary loss of synchronization resulting from a fault in the clock mechanism of one of the parties. Finally, because of the variable and unpredictable nature of network delays, distributed clocks cannot be expected to maintain precise synchronization. Therefore, any timestamp-based pro-cedure must allow for a window of time sufficiently large to accommodate network delays yet sufficiently small to minimize the opportunity for attack.

On the other hand, the challenge-response approach is unsuitable for a con-nectionless type of application, because it requires the overhead of a handshake be-fore any connectionless transmission, effectively negating the chief characteristic of a connectionless transaction. For such applications, reliance on some sort of secure time server and a consistent attempt by each party to keep its clocks in synchroniza-tion may be the best approach (e.g., [LAM92b]).

One-Way Authentication

One application for which encryption is growing in popularity is electronic mail (e-mail). The very nature of electronic mail, and its chief benefit, is that it is not nec-essary for the sender and receiver to be online at the same time. Instead, the e-mail message is forwarded to the receiver’s electronic mailbox, where it is buffered until the receiver is available to read it.

The “envelope” or header of the e-mail message must be in the clear, so that the message can be handled by the store-and-forward e-mail protocol, such as the Simple Mail Transfer Protocol (SMTP) or X.400. However, it is often desirable that the mail-handling protocol not require access to the plaintext form of the message, because that would require trusting the mail-handling mechanism. Accordingly, the e-mail message should be encrypted such that the mail-handling system is not in possession of the decryption key.

A second requirement is that of authentication. Typically, the recipient wants some assurance that the message is from the alleged sender.

15.2 remOte user-authenticatiOn using symmetric encryPtiOn


As was discussed in Chapter 14, a two-level hierarchy of symmetric encryption keys can be used to provide confidentiality for communication in a distributed environ-ment. In general, this strategy involves the use of a trusted key distribution center (KDC). Each party in the network shares a secret key, known as a master key, with the KDC. The KDC is responsible for generating keys to be used for a short time over a connection between two parties, known as session keys, and for distributing those keys using the master keys to protect the distribution. This approach is quite common. As an example, we look at the Kerberos system in Section 15.3. The discus-sion in this subsection is relevant to an understanding of the Kerberos mechanisms.

15.2 / Remote useR-AuthenticAtion using symmetRic encRyPtion 455

Figure 14.3 illustrates a proposal initially put forth by Needham and Schroeder [NEED78] for secret key distribution using a KDC that, as was mentioned in Chapter 14, includes authentication features. The protocol can be summarized as follows.1

1. A S KDC: IDA } IDB }N1

2. KDC S A: E(Ka,[Ks } IDB }N1 }E(Kb,[Ks } IDA])])3. A S B: E(Kb, [Ks } IDA])4. B S A: E(Ks, N2)5. A S B: E(Ks, f(N2))

Secret keys Ka and Kb are shared between A and the KDC and B and the KDC, respectively. The purpose of the protocol is to distribute securely a session key Ks to A and B. A securely acquires a new session key in step 2. The message in step 3 can be decrypted, and hence understood, only by B. Step 4 reflects B’s knowl-edge of Ks, and step 5 assures B of A’s knowledge of Ks and assures B that this is a fresh message because of the use of the nonce N2. Recall from our discussion in Chapter 14 that the purpose of steps 4 and 5 is to prevent a certain type of replay at-tack. In particular, if an opponent is able to capture the message in step 3 and replay it, this might in some fashion disrupt operations at B.

Despite the handshake of steps 4 and 5, the protocol is still vulnerable to a form of replay attack. Suppose that an opponent, X, has been able to compromise an old session key. Admittedly, this is a much more unlikely occurrence than that an opponent has simply observed and recorded step 3. Nevertheless, it is a potential security risk. X can impersonate A and trick B into using the old key by simply replaying step 3. Unless B remembers indefinitely all previous session keys used with A, B will be un-able to determine that this is a replay. If X can intercept the handshake message in step 4, then it can impersonate A’s response in step 5. From this point on, X can send bogus messages to B that appear to B to come from A using an authenticated session key.

Denning [DENN81, DENN82] proposes to overcome this weakness by a modification to the Needham/Schroeder protocol that includes the addition of a timestamp to steps 2 and 3. Her proposal assumes that the master keys, Ka and Kb, are secure, and it consists of the following steps.

1. A S KDC: IDA } IDB

2. KDC S A: E(Ka, [Ks } IDB }T }E(Kb, [Ks } IDA }T])]) 3. A S B: E(Kb, [Ks } IDA }T]) 4. B S A: E(Ks, N1) 5. A S B: E(Ks, f(N1))

T is a timestamp that assures A and B that the session key has only just been generated. Thus, both A and B know that the key distribution is a fresh exchange. A and B can verify timeliness by checking that

0 Clock - T 0 6 ∆t1 + ∆t2

1The portion to the left of the colon indicates the sender and the receiver; the portion to the right indicates the contents of the message; the symbol } indicates concatenation.


where ∆t1 is the estimated normal discrepancy between the KDC’s clock and the local clock (at A or B) and ∆t2 is the expected network delay time. Each node can set its clock against some standard reference source. Because the timestamp T is encrypted using the secure master keys, an opponent, even with knowledge of an old session key, cannot succeed because a replay of step 3 will be detected by B as untimely.

A final point: Steps 4 and 5 were not included in the original presentation [DENN81] but were added later [DENN82]. These steps confirm the receipt of the session key at B.

The Denning protocol seems to provide an increased degree of security com-pared to the Needham/Schroeder protocol. However, a new concern is raised: namely, that this new scheme requires reliance on clocks that are synchronized throughout the network. [GONG92] points out a risk involved. The risk is based on the fact that the distributed clocks can become unsynchronized as a result of sabotage on or faults in the clocks or the synchronization mechanism.2 The problem occurs when a sender’s clock is ahead of the intended recipient’s clock. In this case, an opponent can intercept a message from the sender and replay it later when the timestamp in the message becomes current at the recipient’s site. This replay could cause unexpected results. Gong refers to such attacks as suppress-replay attacks.

One way to counter suppress-replay attacks is to enforce the requirement that parties regularly check their clocks against the KDC’s clock. The other alternative, which avoids the need for clock synchronization, is to rely on handshaking protocols using nonces. This latter alternative is not vulnerable to a suppress-replay attack, because the nonces the recipient will choose in the future are unpredictable to the sender. The Needham/Schroeder protocol relies on nonces only but, as we have seen, has other vulnerabilities.

In [KEHN92], an attempt is made to respond to the concerns about suppress-replay attacks and at the same time fix the problems in the Needham/Schroeder protocol. Subsequently, an inconsistency in this latter protocol was noted and an improved strategy was presented in [NEUM93a].3 The protocol is

1. A S B: IDA }Na

2. B S KDC: IDB }Nb }E(Kb, [IDA }Na }Tb])

3. KDC S A: E(Ka, [IDB }Na }Ks }Tb]) }E(Kb, [IDA }Ks }Tb]) }Nb

4. A S B: E(Kb, [IDA }Ks }Tb]) }E(Ks, Nb)

Let us follow this exchange step by step.

1. A initiates the authentication exchange by generating a nonce, Na, and sending that plus its identifier to B in plaintext. This nonce will be returned to A in an encrypted message that includes the session key, assuring A of its timeliness.

2. B alerts the KDC that a session key is needed. Its message to the KDC in-cludes its identifier and a nonce, Nb. This nonce will be returned to B in an encrypted message that includes the session key, assuring B of its timeliness.

2Such things can and do happen. In recent years, flawed chips were used in a number of computers and other electronic systems to track the time and date. The chips had a tendency to skip forward one day. [NEUM90]3It really is hard to get these things right.

15.2 / Remote useR-AuthenticAtion using symmetRic encRyPtion 457

B’s message to the KDC also includes a block encrypted with the secret key shared by B and the KDC. This block is used to instruct the KDC to issue credentials to A; the block specifies the intended recipient of the credentials, a suggested expiration time for the credentials, and the nonce received from A.

3. The KDC passes on to A B’s nonce and a block encrypted with the secret key that B shares with the KDC. The block serves as a “ticket” that can be used by A for subsequent authentications, as will be seen. The KDC also sends to A a block encrypted with the secret key shared by A and the KDC. This block verifies that B has received A’s initial message (IDB) and that this is a timely message and not a replay (Na), and it provides A with a session key (Ks) and the time limit on its use (Tb).

4. A transmits the ticket to B, together with the B’s nonce, the latter encrypted with the session key. The ticket provides B with the secret key that is used to de-crypt E(Ks, Nb) to recover the nonce. The fact that B’s nonce is encrypted with the session key authenticates that the message came from A and is not a replay.

This protocol provides an effective, secure means for A and B to establish a session with a secure session key. Furthermore, the protocol leaves A in posses-sion of a key that can be used for subsequent authentication to B, avoiding the need to contact the authentication server repeatedly. Suppose that A and B estab-lish a session using the aforementioned protocol and then conclude that session. Subsequently, but within the time limit established by the protocol, A desires a new session with B. The following protocol ensues:

1. A S B: E(Kb, [IDA }Ks }Tb]) }N′a 2. B S A: N′b }E(Ks, N′a) 3. A S B: E(Ks, N′b)

When B receives the message in step 1, it verifies that the ticket has not expired. The newly generated nonces N′a and N′b assure each party that there is no replay attack.

In all the foregoing, the time specified in Tb is a time relative to B’s clock. Thus, this timestamp does not require synchronized clocks, because B checks only self-generated timestamps.


Using symmetric encryption, the decentralized key distribution scenario illustrated in Figure 14.5 is impractical. This scheme requires the sender to issue a request to the intended recipient, await a response that includes a session key, and only then send the message.

With some refinement, the KDC strategy illustrated in Figure 14.3 is a can-didate for encrypted electronic mail. Because we wish to avoid requiring that the recipient (B) be on line at the same time as the sender (A), steps 4 and 5 must be eliminated. For a message with content M, the sequence is as follows:

1. A S KDC: IDA } IDB }N 1

2. KDC S A: E(Ka, [Ks } IDB }N 1 }E(Kb, [Ks } IDA])])

3. A S B: E(Kb, [Ks } IDA]) }E(Ks, M)


This approach guarantees that only the intended recipient of a message will be able to read it. It also provides a level of authentication that the sender is A. As specified, the protocol does not protect against replays. Some measure of defense could be provided by including a timestamp with the message. However, because of the potential delays in the e-mail process, such timestamps may have limited usefulness.

15.3 KerberOs

Kerberos4 is an authentication service developed as part of Project Athena at MIT. The problem that Kerberos addresses is this: Assume an open distributed environ-ment in which users at workstations wish to access services on servers distributed throughout the network. We would like for servers to be able to restrict access to authorized users and to be able to authenticate requests for service. In this envi-ronment, a workstation cannot be trusted to identify its users correctly to network services. In particular, the following three threats exist:

1. A user may gain access to a particular workstation and pretend to be another user operating from that workstation.

2. A user may alter the network address of a workstation so that the requests sent from the altered workstation appear to come from the impersonated workstation.

3. A user may eavesdrop on exchanges and use a replay attack to gain entrance to a server or to disrupt operations.

In any of these cases, an unauthorized user may be able to gain access to services and data that he or she is not authorized to access. Rather than building in elaborate authentication protocols at each server, Kerberos provides a centralized authentica-tion server whose function is to authenticate users to servers and servers to users. Unlike most other authentication schemes described in this book, Kerberos relies exclusively on symmetric encryption, making no use of public-key encryption.

Two versions of Kerberos are in common use. Version 4 [MILL88, STEI88] implementations still exist. Version 5 [KOHL94] corrects some of the security defi-ciencies of version 4 and has been issued as a proposed Internet Standard (RFC 4120 and RFC 4121).5

We begin this section with a brief discussion of the motivation for the Kerberos approach. Then, because of the complexity of Kerberos, it is best to start with a de-scription of the authentication protocol used in version 4. This enables us to see the essence of the Kerberos strategy without considering some of the details required to handle subtle security threats. Finally, we examine version 5.

4“In Greek mythology, a many headed dog, commonly three, perhaps with a serpent’s tail, the guardian of the entrance of Hades.” From Dictionary of Subjects and Symbols in Art, by James Hall, Harper & Row, 1979. Just as the Greek Kerberos has three heads, the modern Kerberos was intended to have three components to guard a network’s gate: authentication, accounting, and audit. The last two heads were never implemented.5Versions 1 through 3 were internal development versions. Version 4 is the “original” Kerberos.

15.3 / KeRbeRos 459

Motivation

If a set of users is provided with dedicated personal computers that have no network connections, then a user’s resources and files can be protected by physically secur-ing each personal computer. When these users instead are served by a centralized time-sharing system, the time-sharing operating system must provide the security. The operating system can enforce access-control policies based on user identity and use the logon procedure to identify users.

Today, neither of these scenarios is typical. More common is a distributed architecture consisting of dedicated user workstations (clients) and distributed or cen-tralized servers. In this environment, three approaches to security can be envisioned.

1. Rely on each individual client workstation to assure the identity of its user or users and rely on each server to enforce a security policy based on user identi-fication (ID).

2. Require that client systems authenticate themselves to servers, but trust the client system concerning the identity of its user.

3. Require the user to prove his or her identity for each service invoked. Also require that servers prove their identity to clients.

In a small, closed environment in which all systems are owned and operated by a single organization, the first or perhaps the second strategy may suffice.6 But in a more open environment in which network connections to other machines are supported, the third approach is needed to protect user information and resources housed at the server. Kerberos supports this third approach. Kerberos assumes a distributed client/server architecture and employs one or more Kerberos servers to provide an authentication service.

The first published report on Kerberos [STEI88] listed the following requirements.

• Secure: A network eavesdropper should not be able to obtain the necessary information to impersonate a user. More generally, Kerberos should be strong enough that a potential opponent does not find it to be the weak link.

• Reliable: For all services that rely on Kerberos for access control, lack of availability of the Kerberos service means lack of availability of the supported services. Hence, Kerberos should be highly reliable and should employ a dis-tributed server architecture with one system able to back up another.

• Transparent: Ideally, the user should not be aware that authentication is tak-ing place beyond the requirement to enter a password.

• Scalable: The system should be capable of supporting large numbers of clients and servers. This suggests a modular, distributed architecture.

To support these requirements, the overall scheme of Kerberos is that of a trusted third-party authentication service that uses a protocol based on that pro-posed by Needham and Schroeder [NEED78], which was discussed in Section 15.2.

6However, even a closed environment faces the threat of attack by a disgruntled employee.


It is trusted in the sense that clients and servers trust Kerberos to mediate their mutual authentication. Assuming the Kerberos protocol is well designed, then the authentication service is secure if the Kerberos server itself is secure.7

Kerberos Version 4

Version 4 of Kerberos makes use of DES, in a rather elaborate protocol, to provide the authentication service. Viewing the protocol as a whole, it is difficult to see the need for the many elements contained therein. Therefore, we adopt a strategy used by Bill Bryant of Project Athena [BRYA88] and build up to the full protocol by look-ing first at several hypothetical dialogues. Each successive dialogue adds additional complexity to counter security vulnerabilities revealed in the preceding dialogue.

After examining the protocol, we look at some other aspects of version 4.

A Simple AuthenticAtion DiAlogue In an unprotected network environment, any client can apply to any server for service. The obvious security risk is that of im-personation. An opponent can pretend to be another client and obtain unauthor-ized privileges on server machines. To counter this threat, servers must be able to confirm the identities of clients who request service. Each server can be required to undertake this task for each client/server interaction, but in an open environment, this places a substantial burden on each server.

An alternative is to use an authentication server (AS) that knows the pass-words of all users and stores these in a centralized database. In addition, the AS shares a unique secret key with each server. These keys have been distributed physi-cally or in some other secure manner. Consider the following hypothetical dialogue:

(1) C S AS: IDC }PC } IDV

(2) AS S C: Ticket

(3) C S V: IDC }Ticket

Ticket = E(Kv, [IDC }ADC } IDV])

where

C = client

AS = authentication server

V = server

IDC = identifier of user on C

IDV = identifier of V

PC = password of user on C

7Remember that the security of the Kerberos server should not automatically be assumed but must be guarded carefully (e.g., in a locked room). It is well to remember the fate of the Greek Kerberos, whom Hercules was ordered by Eurystheus to capture as his Twelfth Labor: “Hercules found the great dog on its chain and seized it by the throat. At once the three heads tried to attack, and Kerberos lashed about with his powerful tail. Hercules hung on grimly, and Kerberos relaxed into unconsciousness. Eurystheus may have been surprised to see Hercules alive—when he saw the three slavering heads and the huge dog they belonged to he was frightened out of his wits, and leapt back into the safety of his great bronze jar.” From The Hamlyn Concise Dictionary of Greek and Roman Mythology, by Michael Stapleton, Hamlyn, 1982.

15.3 / KeRbeRos 461

ADC = network address of C

Kv = secret encryption key shared by AS and V

In this scenario, the user logs on to a workstation and requests access to server V. The client module C in the user’s workstation requests the user’s password and then sends a message to the AS that includes the user’s ID, the server’s ID, and the user’s password. The AS checks its database to see if the user has supplied the proper password for this user ID and whether this user is permitted access to server V. If both tests are passed, the AS accepts the user as authentic and must now convince the server that this user is authentic. To do so, the AS creates a ticket that con-tains the user’s ID and network address and the server’s ID. This ticket is encrypted using the secret key shared by the AS and this server. This ticket is then sent back to C. Because the ticket is encrypted, it cannot be altered by C or by an opponent.

With this ticket, C can now apply to V for service. C sends a message to V con-taining C’s ID and the ticket. V decrypts the ticket and verifies that the user ID in the ticket is the same as the unencrypted user ID in the message. If these two match, the server considers the user authenticated and grants the requested service.

Each of the ingredients of message (3) is significant. The ticket is encrypted to prevent alteration or forgery. The server’s ID (IDV) is included in the ticket so that the server can verify that it has decrypted the ticket properly. IDC is included in the ticket to indicate that this ticket has been issued on behalf of C. Finally, ADC serves to counter the following threat. An opponent could capture the ticket transmitted in message (2), then use the name IDC and transmit a message of form (3) from another workstation. The server would receive a valid ticket that matches the user ID and grant access to the user on that other workstation. To prevent this attack, the AS includes in the ticket the network address from which the original request came. Now the ticket is valid only if it is transmitted from the same workstation that initially requested the ticket.

A more Secure AuthenticAtion DiAlogue Although the foregoing scenario solves some of the problems of authentication in an open network environment, problems remain. Two in particular stand out. First, we would like to minimize the number of times that a user has to enter a password. Suppose each ticket can be used only once. If user C logs on to a workstation in the morning and wishes to check his or her mail at a mail server, C must supply a password to get a ticket for the mail server. If C wishes to check the mail several times during the day, each attempt requires reentering the password. We can improve matters by saying that tickets are reusable. For a single logon session, the workstation can store the mail server ticket after it is received and use it on behalf of the user for multiple accesses to the mail server.

However, under this scheme, it remains the case that a user would need a new ticket for every different service. If a user wished to access a print server, a mail server, a file server, and so on, the first instance of each access would require a new ticket and hence require the user to enter the password.

The second problem is that the earlier scenario involved a plaintext transmis-sion of the password [message (1)]. An eavesdropper could capture the password and use any service accessible to the victim.


To solve these additional problems, we introduce a scheme for avoiding plain-text passwords and a new server, known as the ticket-granting server (TGS). The new (but still hypothetical) scenario is as follows.

Once per user logon session:

(1) C S AS: IDC } IDtgs

(2) AS S C: E(Kc, Tickettgs)

Once per type of service:

(3) C S TGS: IDC } IDV }Tickettgs

(4) TGS S C: Ticketv

Once per service session:

(5) C S V: IDC }Ticketv

Tickettgs = E(Ktgs, [IDC }ADC } IDtgs }TS1 }Lifetime1])Ticketv = E(Kv, [IDC } ADC } IDv }TS2 }Lifetime2])

The new service, TGS, issues tickets to users who have been authenticated to AS. Thus, the user first requests a ticket-granting ticket (Tickettgs) from the AS. The client module in the user workstation saves this ticket. Each time the user requires access to a new service, the client applies to the TGS, using the ticket to authenti-cate itself. The TGS then grants a ticket for the particular service. The client saves each service-granting ticket and uses it to authenticate its user to a server each time a particular service is requested. Let us look at the details of this scheme:

1. The client requests a ticket-granting ticket on behalf of the user by sending its user’s ID to the AS, together with the TGS ID, indicating a request to use the TGS service.

2. The AS responds with a ticket that is encrypted with a key that is derived from the user’s password (Kc), which is already stored at the AS. When this response arrives at the client, the client prompts the user for his or her pass-word, generates the key, and attempts to decrypt the incoming message. If the correct password is supplied, the ticket is successfully recovered.

Because only the correct user should know the password, only the correct user can recover the ticket. Thus, we have used the password to obtain credentials from Kerberos without having to transmit the password in plaintext. The ticket itself consists of the ID and network address of the user, and the ID of the TGS. This corresponds to the first scenario. The idea is that the client can use this ticket to request multiple service-granting tickets. So the ticket-granting ticket is to be reus-able. However, we do not wish an opponent to be able to capture the ticket and use it. Consider the following scenario: An opponent captures the login ticket and waits until the user has logged off his or her workstation. Then the opponent either gains access to that workstation or configures his workstation with the same net-work address as that of the victim. The opponent would be able to reuse the ticket

15.3 / KeRbeRos 463

to spoof the TGS. To counter this, the ticket includes a timestamp, indicating the date and time at which the ticket was issued, and a lifetime, indicating the length of time for which the ticket is valid (e.g., eight hours). Thus, the client now has a reusable ticket and need not bother the user for a password for each new service request. Finally, note that the ticket-granting ticket is encrypted with a secret key known only to the AS and the TGS. This prevents alteration of the ticket. The ticket is reencrypted with a key based on the user’s password. This assures that the ticket can be recovered only by the correct user, providing the authentication.

Now that the client has a ticket-granting ticket, access to any server can be obtained with steps 3 and 4.

3. The client requests a service-granting ticket on behalf of the user. For this pur-pose, the client transmits a message to the TGS containing the user’s ID, the ID of the desired service, and the ticket-granting ticket.

4. The TGS decrypts the incoming ticket using a key shared only by the AS and the TGS (Ktgs) and verifies the success of the decryption by the presence of its ID. It checks to make sure that the lifetime has not expired. Then it compares the user ID and network address with the incoming information to authenti-cate the user. If the user is permitted access to the server V, the TGS issues a ticket to grant access to the requested service.

The service-granting ticket has the same structure as the ticket-granting ticket. Indeed, because the TGS is a server, we would expect that the same elements are needed to authenticate a client to the TGS and to authenticate a client to an appli-cation server. Again, the ticket contains a timestamp and lifetime. If the user wants access to the same service at a later time, the client can simply use the previously acquired service-granting ticket and need not bother the user for a password. Note that the ticket is encrypted with a secret key (Kv) known only to the TGS and the server, preventing alteration.

Finally, with a particular service-granting ticket, the client can gain access to the corresponding service with step 5.

5. The client requests access to a service on behalf of the user. For this purpose, the client transmits a message to the server containing the user’s ID and the service-granting ticket. The server authenticates by using the contents of the ticket.

This new scenario satisfies the two requirements of only one password query per user session and protection of the user password.

the VerSion 4 AuthenticAtion DiAlogue Although the foregoing scenario en-hances security compared to the first attempt, two additional problems remain. The heart of the first problem is the lifetime associated with the ticket-granting ticket. If this lifetime is very short (e.g., minutes), then the user will be repeatedly asked for a password. If the lifetime is long (e.g., hours), then an opponent has a greater opportunity for replay. An opponent could eavesdrop on the network and capture a copy of the ticket-granting ticket and then wait for the legitimate user to log out. Then the opponent could forge the legitimate user’s network address and send the message of step (3) to the TGS. This would give the opponent unlimited access to the resources and files available to the legitimate user.


Similarly, if an opponent captures a service-granting ticket and uses it before it expires, the opponent has access to the corresponding service.

Thus, we arrive at an additional requirement. A network service (the TGS or an application service) must be able to prove that the person using a ticket is the same person to whom that ticket was issued.

The second problem is that there may be a requirement for servers to authenticate themselves to users. Without such authentication, an opponent could sabotage the configuration so that messages to a server were directed to another location. The false server would then be in a position to act as a real server and capture any information from the user and deny the true service to the user.

We examine these problems in turn and refer to Table 15.1, which shows the actual Kerberos protocol. Figure 15.1 provides a simplified overview.

First, consider the problem of captured ticket-granting tickets and the need to determine that the ticket presenter is the same as the client for whom the ticket was issued. The threat is that an opponent will steal the ticket and use it before it expires. To get around this problem, let us have the AS provide both the client and the TGS with a secret piece of information in a secure manner. Then the client can prove its identity to the TGS by revealing the secret information—again in a secure manner. An efficient way of accomplishing this is to use an encryption key as the secure information; this is referred to as a session key in Kerberos.

Table 15.1a shows the technique for distributing the session key. As before, the client sends a message to the AS requesting access to the TGS. The AS responds with a message, encrypted with a key derived from the user’s password (Kc), that contains the ticket. The encrypted message also contains a copy of the session key, Kc,tgs, where the subscripts indicate that this is a session key for C and TGS. Because this session key is inside the message encrypted with Kc, only the user’s client can read it. The same session key is included in the ticket, which can be read only by the TGS. Thus, the session key has been securely delivered to both C and the TGS.

Table 15.1 Summary of Kerberos Version 4 Message Exchanges

(1) C S AS IDc } IDtgs }TS1

(2) AS S C E(Kc, [Kc, tgs } IDtgs }TS2 � � Lifetime2 }Tickettgs])

Tickettgs = E(Ktgs, [Kc, tgs } IDC }ADC } IDtgs }TS2 }Lifetime2])

(a) Authentication Service Exchange to obtain ticket-granting ticket

(3) C S TGS IDv }Tickettgs }Authenticatorc

(4) TGS S C E(Kc, tgs, [Kc, v } IDv }TS4 }Ticketv])

Tickettgs = E(Ktgs, [Kc, tgs } IDC }ADC } IDtgs }TS2 }Lifetime2])

Ticketv = E(Kv, [Kc, v } IDC }ADC } IDv }TS4 }Lifetime4])

Authenticatorc = E(Kc, tgs, [IDC }ADC }TS3])

(b) Ticket-Granting Service Exchange to obtain service-granting ticket

(5) C S V Ticketv }Authenticatorc

(6) V S C E(Kc,v, [TS5 + 1]) (for mutual authentication)

Ticketv = E(Kv, [Kc, v } IDC }ADC } IDv }TS4 }Lifetime4])

Authenticatorc = E (Kc, v, [IDC }ADC }TS5])

(c) Client/Server Authentication Exchange to obtain service

15.3 / KeRbeRos 465

Note that several additional pieces of information have been added to this first phase of the dialogue. Message (1) includes a timestamp, so that the AS knows that the message is timely. Message (2) includes several elements of the ticket in a form accessible to C. This enables C to confirm that this ticket is for the TGS and to learn its expiration time.

Armed with the ticket and the session key, C is ready to approach the TGS. As before, C sends the TGS a message that includes the ticket plus the ID of the re-quested service [message (3) in Table 15.1b]. In addition, C transmits an authentica-tor, which includes the ID and address of C’s user and a timestamp. Unlike the ticket, which is reusable, the authenticator is intended for use only once and has a very short lifetime. The TGS can decrypt the ticket with the key that it shares with the AS. This ticket indicates that user C has been provided with the session key Kc,tgs. In effect, the ticket says, “Anyone who uses Kc,tgs must be C.” The TGS uses the session key to decrypt the authenticator. The TGS can then check the name and address from the authenticator with that of the ticket and with the network address of the incoming message. If all match, then the TGS is assured that the sender of the ticket is indeed the ticket’s real owner. In effect, the authenticator says, “At time TS3, I hereby use Kc,tgs.” Note that the ticket does not prove anyone’s identity but is a way to distribute

Authenticationserver

Ticket-granting

server (TGS)

Host/application

server

request ticket-

granting ticket

once peruser logonsession

1. User logs on toworkstation andrequests service on host

3. Workstation promptsuser for password to decryptincoming message, and thensend ticket andauthenticator that containsuser’s name, networkaddress, and time to TGS.

ticket + session key

request service-

granting ticket

ticket + session key

once pertype of service

4. TGS decrypts ticket andauthenticator, verifies request,and then creates ticket forrequested application server.

Kerberos

5. Workstation sendsticket and authenticatorto host.

6. Host verifies thatticket and authenticatormatch, and then grantsaccess to service. Ifmutual authentication isrequired, server returnsan authenticator.

request service

provide server

authenticatoronce perservice session

2. AS verifies user’s access right indatabase, and creates ticket-granting ticketand session key. Results are encryptedusing key derived from user’s password.

Figure 15.1 Overview of Kerberos


keys securely. It is the authenticator that proves the client’s identity. Because the au-thenticator can be used only once and has a short lifetime, the threat of an opponent stealing both the ticket and the authenticator for presentation later is countered.

The reply from the TGS in message (4) follows the form of message (2). The message is encrypted with the session key shared by the TGS and C and includes a session key to be shared between C and the server V, the ID of V, and the time-stamp of the ticket. The ticket itself includes the same session key.

C now has a reusable service-granting ticket for V. When C presents this ticket, as shown in message (5), it also sends an authenticator. The server can decrypt the ticket, recover the session key, and decrypt the authenticator.

If mutual authentication is required, the server can reply as shown in message (6) of Table 15.1. The server returns the value of the timestamp from the authenti-cator, incremented by 1, and encrypted in the session key. C can decrypt this mes-sage to recover the incremented timestamp. Because the message was encrypted by the session key, C is assured that it could have been created only by V. The contents of the message assure C that this is not a replay of an old reply.

Finally, at the conclusion of this process, the client and server share a secret key. This key can be used to encrypt future messages between the two or to ex-change a new random session key for that purpose.

Figure 15.2 illustrates the Kerberos exchanges among the parties. Table 15.2 summarizes the justification for each of the elements in the Kerberos protocol.

Client

Client authenticationIDc || IDtgs || TS1

Tickettgs, server ID, and client authenticationIDv || Tickettgs || Authenticatorc

Shared key and ticketE(Kc,tgs, [Kc,v || IDv || TS4 || Ticketv])

Ticketv and client authenticationTicketv || Authenticatorc

Service grantedE(Kc,v, [TS5 + 1])

Shared key and ticketE(Kc, [Kc, tgs || IDtgs || TS2 ||

Lifetime2 || Tickettgs])

Authenticationserver (AS)

Ticket-grantingserver (TGS)

Serviceprovider

Figure 15.2 Kerberos Exchanges

15.3 / KeRbeRos 467

Table 15.2 Rationale for the Elements of the Kerberos Version 4 Protocol

Message (1) Client requests ticket-granting ticket.

IDC Tells AS identity of user from this client.

IDtgs Tells AS that user requests access to TGS.

TS1 Allows AS to verify that client’s clock is synchronized with that of AS.

Message (2) AS returns ticket-granting ticket.

Kc Encryption is based on user’s password, enabling AS and client to verify password, and protecting contents of message (2).

Kc, tgs Copy of session key accessible to client created by AS to permit secure exchange between client and TGS without requiring them to share a permanent key.

IDtgs Confirms that this ticket is for the TGS.

TS2 Informs client of time this ticket was issued.

Lifetime2 Informs client of the lifetime of this ticket.

Tickettgs Ticket to be used by client to access TGS.

(a) Authentication Service Exchange

Message (3) Client requests service-granting ticket.

IDV Tells TGS that user requests access to server V.

Tickettgs Assures TGS that this user has been authenticated by AS.

Authenticatorc Generated by client to validate ticket.

Message (4) TGS returns service-granting ticket.

Kc, tgs Key shared only by C and TGS protects contents of message (4).

Kc, v Copy of session key accessible to client created by TGS to permit secure exchange between client and server without requiring them to share a permanent key.

IDV Confirms that this ticket is for server V.

TS4 Informs client of time this ticket was issued.

TicketV Ticket to be used by client to access server V.

Tickettgs Reusable so that user does not have to reenter password.

Ktgs Ticket is encrypted with key known only to AS and TGS, to prevent tampering.

Kc, tgs Copy of session key accessible to TGS used to decrypt authenticator, thereby authenticating ticket.

IDC Indicates the rightful owner of this ticket.

ADC Prevents use of ticket from workstation other than one that initially requested the ticket.

IDtgs Assures server that it has decrypted ticket properly.

TS2 Informs TGS of time this ticket was issued.

Lifetime2 Prevents replay after ticket has expired.

Authenticatorc Assures TGS that the ticket presenter is the same as the client for whom the ticket was issued has very short lifetime to prevent replay.

Kc,tgs Authenticator is encrypted with key known only to client and TGS, to prevent tampering.

IDC Must match ID in ticket to authenticate ticket.

ADC Must match address in ticket to authenticate ticket.

TS3 Informs TGS of time this authenticator was generated.

(b) Ticket-Granting Service Exchange

(continued)


Message (5) Client requests service.

TicketV Assures server that this user has been authenticated by AS.

Authenticatorc Generated by client to validate ticket.

Message (6) Optional authentication of server to client.

Kc, v Assures C that this message is from V.

TS5 + 1 Assures C that this is not a replay of an old reply.

Ticketv Reusable so that client does not need to request a new ticket from TGS for each access to the same server.

Kv Ticket is encrypted with key known only to TGS and server, to prevent tampering.

Kc, v Copy of session key accessible to client; used to decrypt authenticator, thereby authenticating ticket.

IDC Indicates the rightful owner of this ticket.

ADC Prevents use of ticket from workstation other than one that initially requested the ticket.

IDV Assures server that it has decrypted ticket properly.

TS4 Informs server of time this ticket was issued.

Lifetime4 Prevents replay after ticket has expired.

Authenticatorc Assures server that the ticket presenter is the same as the client for whom the ticket was issued; has very short lifetime to prevent replay.

Kc, v Authenticator is encrypted with key known only to client and server, to prevent tampering.

IDC Must match ID in ticket to authenticate ticket.

ADC Must match address in ticket to authenticate ticket.

TS5 Informs server of time this authenticator was generated.

(c) Client/Server Authentication Exchange

KerberoS reAlmS AnD multiple Kerberi A full-service Kerberos environment consisting of a Kerberos server, a number of clients, and a number of application servers requires the following:

1. The Kerberos server must have the user ID and hashed passwords of all partic-ipating users in its database. All users are registered with the Kerberos server.

2. The Kerberos server must share a secret key with each server. All servers are registered with the Kerberos server.

Such an environment is referred to as a Kerberos realm. The concept of realm can be explained as follows. A Kerberos realm is a set of managed nodes that share the same Kerberos database. The Kerberos database resides on the Kerberos master computer system, which should be kept in a physically secure room. A read-only copy of the Kerberos database might also reside on other Kerberos computer systems. However, all changes to the database must be made on the master com-puter system. Changing or accessing the contents of a Kerberos database requires the Kerberos master password. A related concept is that of a Kerberos principal, which is a service or user that is known to the Kerberos system. Each Kerberos

Table 15.2 Continued

15.3 / KeRbeRos 469

principal is identified by its principal name. Principal names consist of three parts: a service or user name, an instance name, and a realm name.

Networks of clients and servers under different administrative organizations typically constitute different realms. That is, it generally is not practical or does not conform to administrative policy to have users and servers in one administrative domain registered with a Kerberos server elsewhere. However, users in one realm may need access to servers in other realms, and some servers may be willing to pro-vide service to users from other realms, provided that those users are authenticated.

Kerberos provides a mechanism for supporting such interrealm authentication. For two realms to support interrealm authentication, a third requirement is added:

3. The Kerberos server in each interoperating realm shares a secret key with the server in the other realm. The two Kerberos servers are registered with each other.

The scheme requires that the Kerberos server in one realm trust the Kerberos server in the other realm to authenticate its users. Furthermore, the participating servers in the second realm must also be willing to trust the Kerberos server in the first realm.

With these ground rules in place, we can describe the mechanism as follows (Figure 15.3): A user wishing service on a server in another realm needs a ticket for that server. The user’s client follows the usual procedures to gain access to the local TGS and then requests a ticket-granting ticket for a remote TGS (TGS in another realm). The client can then apply to the remote TGS for a service-granting ticket for the desired server in the realm of the remote TGS.

The details of the exchanges illustrated in Figure 15.3 are as follows (compare Table 15.1).

(1) C S AS: IDc } IDtgs }TS1

(2) AS S C: E(Kc, [Kc, tgs } IDtgs }TS2 }Lifetime2 }Tickettgs])

(3) C S TGS: IDtgsrem }Tickettgs }Authenticatorc

(4) TGS S C: E(Kc,tgs, [Kc, tgsrem } IDtgsrem }TS4 }Tickettgsrem])

(5) C S TGSrem: IDvrem }Tickettgsrem }Authenticatorc

(6) TGSrem S C: E(Kc,tgsrem, [Kc, vrem } IDvrem }TS6 }Ticketvrem])

(7) C S Vrem: Ticketvrem }Authenticatorc

The ticket presented to the remote server (Vrem) indicates the realm in which the user was originally authenticated. The server chooses whether to honor the re-mote request.

One problem presented by the foregoing approach is that it does not scale well to many realms. If there are N realms, then there must be N(N - 1)/2 secure key exchanges so that each Kerberos realm can interoperate with all other Kerberos realms.

Kerberos Version 5

Kerberos version 5 is specified in RFC 4120 and provides a number of improve-ments over version 4 [KOHL94]. To begin, we provide an overview of the changes from version 4 to version 5 and then look at the version 5 protocol.


DifferenceS between VerSionS 4 AnD 5 Version 5 is intended to address the limita-tions of version 4 in two areas: environmental shortcomings and technical deficien-cies. Let us briefly summarize the improvements in each area.8

Kerberos version 4 was developed for use within the Project Athena environ-ment and, accordingly, did not fully address the need to be of general purpose. This led to the following environmental shortcomings.

1. Encryption system dependence: Version 4 requires the use of DES. Export restriction on DES as well as doubts about the strength of DES were thus of

Kerberos

Client

Realm A

AS

TGSServer

Realm B

1. request ticket for local TGS

2. ticket for local TGS

3. request ticket for remote TGS

4. ticket for remote TGS

5. request ticket for remote server

6. ticket for remote server

7. request remote service

Kerberos

AS

TGS

Figure 15.3 Request for Service in Another Realm

8The following discussion follows the presentation in [KOHL94].

15.3 / KeRbeRos 471

concern. In version 5, ciphertext is tagged with an encryption-type identifier so that any encryption technique may be used. Encryption keys are tagged with a type and a length, allowing the same key to be used in different algorithms and allowing the specification of different variations on a given algorithm.

2. Internet protocol dependence: Version 4 requires the use of Internet Protocol (IP) addresses. Other address types, such as the ISO network address, are not accommodated. Version 5 network addresses are tagged with type and length, allowing any network address type to be used.

3. Message byte ordering: In version 4, the sender of a message employs a byte ordering of its own choosing and tags the message to indicate least significant byte in lowest address or most significant byte in lowest address. This tech-niques works but does not follow established conventions. In version 5, all mes-sage structures are defined using Abstract Syntax Notation One (ASN.1) and Basic Encoding Rules (BER), which provide an unambiguous byte ordering.

4. Ticket lifetime: Lifetime values in version 4 are encoded in an 8-bit quantity in units of five minutes. Thus, the maximum lifetime that can be expressed is 28 * 5 = 1280 minutes (a little over 21 hours). This may be inadequate for some applications (e.g., a long-running simulation that requires valid Kerberos credentials throughout execution). In version 5, tickets include an explicit start time and end time, allowing tickets with arbitrary lifetimes.

5. Authentication forwarding: Version 4 does not allow credentials issued to one client to be forwarded to some other host and used by some other client. This capability would enable a client to access a server and have that server access another server on behalf of the client. For example, a client issues a request to a print server that then accesses the client’s file from a file server, using the cli-ent’s credentials for access. Version 5 provides this capability.

6. Interrealm authentication: In version 4, interoperability among N realms requires on the order of N 2 Kerberos-to-Kerberos relationships, as described earlier. Version 5 supports a method that requires fewer relationships, as described shortly.

Apart from these environmental limitations, there are technical deficiencies in the version 4 protocol itself. Most of these deficiencies were documented in [BELL90], and version 5 attempts to address these. The deficiencies are the following.

1. Double encryption: Note in Table 15.1 [messages (2) and (4)] that tickets pro-vided to clients are encrypted twice—once with the secret key of the target server and then again with a secret key known to the client. The second en-cryption is not necessary and is computationally wasteful.

2. PCBC encryption: Encryption in version 4 makes use of a nonstandard mode of DES known as propagating cipher block chaining (PCBC).9 It has been demon-strated that this mode is vulnerable to an attack involving the interchange of ci-phertext blocks [KOHL89]. PCBC was intended to provide an integrity check as part of the encryption operation. Version 5 provides explicit integrity mechanisms,

9This is described in Appendix T.


allowing the standard CBC mode to be used for encryption. In particular, a check-sum or hash code is attached to the message prior to encryption using CBC.

3. Session keys: Each ticket includes a session key that is used by the client to encrypt the authenticator sent to the service associated with that ticket. In ad-dition, the session key may subsequently be used by the client and the server to protect messages passed during that session. However, because the same ticket may be used repeatedly to gain service from a particular server, there is the risk that an opponent will replay messages from an old session to the client or the server. In version 5, it is possible for a client and server to negotiate a subsession key, which is to be used only for that one connection. A new access by the client would result in the use of a new subsession key.

4. Password attacks: Both versions are vulnerable to a password attack. The mes-sage from the AS to the client includes material encrypted with a key based on the client’s password.10 An opponent can capture this message and attempt to decrypt it by trying various passwords. If the result of a test decryption is of the proper form, then the opponent has discovered the client’s password and may subsequently use it to gain authentication credentials from Kerberos. This is the same type of password attack described in Chapter 21, with the same kinds of countermeasures being applicable. Version 5 does provide a mechanism known as preauthentication, which should make password attacks more difficult, but it does not prevent them.

the VerSion 5 AuthenticAtion DiAlogue Table 15.3 summarizes the basic ver-sion 5 dialogue. This is best explained by comparison with version 4 (Table 15.1).

10Appendix T describes the mapping of passwords to encryption keys.

Table 15.3 Summary of Kerberos Version 5 Message Exchanges

(1) C S AS Options } IDc }Realmc } IDtgs }Times }Nonce1

(2) AS S C RealmC } IDC }Tickettgs }E(Kc, [Kc,tgs }Times }Nonce1 }Realmtgs } IDtgs])

Tickettgs = E(Ktgs, [Flags }Kc,tgs }Realmc } IDC }ADC }Times])

(a) Authentication Service Exchange to obtain ticket-granting ticket

(3) C S TGS Options } IDv }Times }Nonce2 }Tickettgs }Authenticatorc

(4) TGS S C Realmc } IDC }Ticketv }E(Kc,tgs, [Kc,v }Times }Nonce2 }Realmv } IDv])

Tickettgs = E(Ktgs, [Flags }Kc,tgs }Realmc } IDC }ADC }Times])

Ticketv = E(Kv, [Flags }Kc,v }Realmc } IDC }ADC }Times])

Authenticatorc = E(Kc,tgs, [IDC }Realmc }TS1])

(b) Ticket-Granting Service Exchange to obtain service-granting ticket

(5) C S V Options }Ticketv }Authenticatorc

(6) V S C EKc, v [TS2 }Subkey }Seq #]

Ticketv = E(Kv, [Flag }Kc,v }Realmc } IDC }ADC }Times])

Authenticatorc = E(Kc,v, [IDC }Relamc }TS2 }Subkey }Seq #])

(c) Client/Server Authentication Exchange to obtain service

15.3 / KeRbeRos 473

First, consider the authentication service exchange. Message (1) is a client re-quest for a ticket-granting ticket. As before, it includes the ID of the user and the TGS. The following new elements are added:

• Realm: Indicates realm of user

• Options: Used to request that certain flags be set in the returned ticket

• Times: Used by the client to request the following time settings in the ticket:

—from: the desired start time for the requested ticket—till: the requested expiration time for the requested ticket—rtime: requested renew-till time

• Nonce: A random value to be repeated in message (2) to assure that the re-sponse is fresh and has not been replayed by an opponent

Message (2) returns a ticket-granting ticket, identifying information for the client, and a block encrypted using the encryption key based on the user’s password. This block includes the session key to be used between the client and the TGS, times specified in message (1), the nonce from message (1), and TGS identifying informa-tion. The ticket itself includes the session key, identifying information for the client, the requested time values, and flags that reflect the status of this ticket and the requested options. These flags introduce significant new functionality to version 5. For now, we defer a discussion of these flags and concentrate on the overall struc-ture of the version 5 protocol.

Let us now compare the ticket-granting service exchange for versions 4 and 5. We see that message (3) for both versions includes an authenticator, a ticket, and the name of the requested service. In addition, version 5 includes requested times and options for the ticket and a nonce—all with functions similar to those of mes-sage (1). The authenticator itself is essentially the same as the one used in version 4.

Message (4) has the same structure as message (2). It returns a ticket plus information needed by the client, with the information encrypted using the session key now shared by the client and the TGS.

Finally, for the client/server authentication exchange, several new features ap-pear in version 5. In message (5), the client may request as an option that mutual authentication is required. The authenticator includes several new fields:

• Subkey: The client’s choice for an encryption key to be used to protect this specific application session. If this field is omitted, the session key from the ticket (Kc,v) is used.

• Sequence number: An optional field that specifies the starting sequence num-ber to be used by the server for messages sent to the client during this session. Messages may be sequence numbered to detect replays.

If mutual authentication is required, the server responds with message (6). This message includes the timestamp from the authenticator. Note that in version 4, the timestamp was incremented by one. This is not necessary in version 5, because the nature of the format of messages is such that it is not possible for an oppo-nent to create message (6) without knowledge of the appropriate encryption keys. The subkey field, if present, overrides the subkey field, if present, in message (5).


The optional sequence number field specifies the starting sequence number to be used by the client.

ticKet flAgS The flags field included in tickets in version 5 supports expanded functionality compared to that available in version 4. Table 15.4 summarizes the flags that may be included in a ticket.

The INITIAL flag indicates that this ticket was issued by the AS, not by the TGS. When a client requests a service-granting ticket from the TGS, it presents a ticket-granting ticket obtained from the AS. In version 4, this was the only way to obtain a service-granting ticket. Version 5 provides the additional capability that the client can get a service-granting ticket directly from the AS. The utility of this is as follows: A server, such as a password-changing server, may wish to know that the client’s password was recently tested.

The PRE-AUTHENT flag, if set, indicates that when the AS received the initial request [message (1)], it authenticated the client before issuing a ticket. The exact form of this preauthentication is left unspecified. As an example, the MIT implementation of version 5 has encrypted timestamp preauthentication, enabled by default. When a user wants to get a ticket, it has to send to the AS a preauthentication block containing a random confounder, a version number, and a timestamp all encrypted in the client’s password-based key. The AS decrypts the block and will not send a ticket-granting ticket back unless the timestamp in the preauthentication block is within the allowable time skew (time interval to account for clock drift and network delays). Another possibility is the use of a smart card that generates continually changing passwords that are included in the preauthenticated messages. The passwords generated by the card can be

Table 15.4 Kerberos Version 5 Flags

INITIAL This ticket was issued using the AS protocol and not issued based on a ticket-granting ticket.

PRE-AUTHENT During initial authentication, the client was authenticated by the KDC before a ticket was issued.

HW-AUTHENT The protocol employed for initial authentication required the use of hardware expected to be possessed solely by the named client.

RENEWABLE Tells TGS that this ticket can be used to obtain a replacement ticket that expires at a later date.

MAY-POSTDATE Tells TGS that a postdated ticket may be issued based on this ticket-granting ticket.

POSTDATED Indicates that this ticket has been postdated; the end server can check the authtime field to see when the original authentication occurred.

INVALID This ticket is invalid and must be validated by the KDC before use.

PROXIABLE Tells TGS that a new service-granting ticket with a different network address may be issued based on the presented ticket.

PROXY Indicates that this ticket is a proxy.

FORWARDABLE Tells TGS that a new ticket-granting ticket with a different network address may be issued based on this ticket-granting ticket.

FORWARDED Indicates that this ticket has either been forwarded or was issued based on authentication involving a forwarded ticket-granting ticket.

15.3 / KeRbeRos 475

based on a user’s password but be transformed by the card so that, in effect, arbitrary passwords are used. This prevents an attack based on easily guessed passwords. If a smart card or similar device was used, this is indicated by the HW-AUTHENT flag.

When a ticket has a long lifetime, there is the potential for it to be stolen and used by an opponent for a considerable period. If a short lifetime is used to lessen the threat, then overhead is involved in acquiring new tickets. In the case of a ticket-granting ticket, the client would either have to store the user’s secret key, which is clearly risky, or repeatedly ask the user for a password. A compromise scheme is the use of renewable tickets. A ticket with the RENEWABLE flag set includes two ex-piration times: One for this specific ticket and one that is the latest permissible value for an expiration time. A client can have the ticket renewed by presenting it to the TGS with a requested new expiration time. If the new time is within the limit of the latest permissible value, the TGS can issue a new ticket with a new session time and a later specific expiration time. The advantage of this mechanism is that the TGS may refuse to renew a ticket reported as stolen.

A client may request that the AS provide a ticket-granting ticket with the MAY-POSTDATE flag set. The client can then use this ticket to request a ticket that is flagged as POSTDATED and INVALID from the TGS. Subsequently, the client may submit the postdated ticket for validation. This scheme can be useful for running a long batch job on a server that requires a ticket periodically. The client can obtain a number of tickets for this session at once, with spread out time values. All but the first ticket are initially invalid. When the execution reaches a point in time when a new ticket is required, the client can get the appropriate ticket validated. With this approach, the client does not have to repeatedly use its ticket-granting ticket to obtain a service-granting ticket.

In version 5, it is possible for a server to act as a proxy on behalf of a client, in effect adopting the credentials and privileges of the client to request a service from another server. If a client wishes to use this mechanism, it requests a ticket-granting ticket with the PROXIABLE flag set. When this ticket is presented to the TGS, the TGS is permitted to issue a service-granting ticket with a different network address; this latter ticket will have its PROXY flag set. An application re-ceiving such a ticket may accept it or require additional authentication to provide an audit trail.11

The proxy concept is a limited case of the more powerful forwarding pro-cedure. If a ticket is set with the FORWARDABLE flag, a TGS can issue to the requestor a ticket-granting ticket with a different network address and the FORWARDED flag set. This ticket then can be presented to a remote TGS. This capability allows a client to gain access to a server on another realm without re-quiring that each Kerberos maintain a secret key with Kerberos servers in every other realm. For example, realms could be structured hierarchically. Then a client could walk up the tree to a common node and then back down to reach a target realm. Each step of the walk would involve forwarding a ticket-granting ticket to the next TGS in the path.

11For a discussion of some of the possible uses of the proxy capability, see [NEUM93b].


15.4 remOte user authenticatiOn using asymmetric encryPtiOn


In Chapter 14, we presented one approach to the use of public-key encryption for the purpose of session-key distribution (Figure 14.9). This protocol assumes that each of the two parties is in possession of the current public key of the other. It may not be practical to require this assumption.

A protocol using timestamps is provided in [DENN81]:

1. A S AS: IDA } IDB

2. AS S A: E(PRas, [IDA }PUa }T]) }E(PRas, [IDB }PUb }T])

3. A S B: E(PRas, [IDA }PUa }T]) }E(PRas, [IDB }PUb }T]) }

E(PUb, E(PRa, [Ks }T]))

In this case, the central system is referred to as an authentication server (AS), because it is not actually responsible for secret-key distribution. Rather, the AS pro-vides public-key certificates. The session key is chosen and encrypted by A; hence, there is no risk of exposure by the AS. The timestamps protect against replays of compromised keys.

This protocol is compact but, as before, requires the synchronization of clocks. Another approach, proposed by Woo and Lam [WOO92a], makes use of nonces. The protocol consists of the following steps.


2. KDC S A: E(PRauth, [IDB }PUb])

3. A S B: E(PUb, [Na } IDA])

4. B S KDC: IDA } IDB }E(PUauth, Na)

5. KDC S B: E(PRauth, [IDA }PUa]) }E(PUb, E(PRauth, [Na }Ks } IDB]))

6. B S A: E(PUa, [E(PRauth, [(Na }Ks } IDB)]) }Nb])

7. A S B: E(Ks, Nb)

In step 1, A informs the KDC of its intention to establish a secure connection with B. The KDC returns to A a copy of B’s public-key certificate (step 2). Using B’s public key, A informs B of its desire to communicate and sends a nonce Na (step 3). In step 4, B asks the KDC for A’s public-key certificate and requests a session key; B includes A’s nonce so that the KDC can stamp the session key with that nonce. The nonce is protected using the KDC’s public key. In step 5, the KDC returns to B a copy of A’s public-key certificate, plus the information {Na, Ks, IDB}. This infor-mation basically says that Ks is a secret key generated by the KDC on behalf of B and tied to Na; the binding of Ks and Na will assure A that Ks is fresh. This triple is encrypted using the KDC’s private key to allow B to verify that the triple is in fact from the KDC. It is also encrypted using B’s public key so that no other entity may

15.4 / Remote useR AuthenticAtion using AsymmetRic encRyPtion 477

use the triple in an attempt to establish a fraudulent connection with A. In step 6, the triple {Na, Ks, IDB}, still encrypted with the KDC’s private key, is relayed to A, together with a nonce Nb generated by B. All the foregoing are encrypted using A’s public key. A retrieves the session key Ks, uses it to encrypt Nb, and returns it to B. This last message assures B of A’s knowledge of the session key.

This seems to be a secure protocol that takes into account the various attacks. However, the authors themselves spotted a flaw and submitted a revised version of the algorithm in [WOO92b]:


2. KDC S A: E(PRauth, [IDB }PUb])

3. A S B: E(PUb, [Na } IDA])

4. B S KDC: IDA } IDB }E(PUauth, Na)

5. KDC S B: E(PRauth, [IDA } PUa]) }E(PUb, E(PRauth, [Na }Ks } IDA } IDB]))

6. B S A: E(PUa, [Nb }E(PRauth, [Na }Ks } IDA } IDB])])

7. A S B: E(Ks, Nb)

The identifier of A, IDA, is added to the set of items encrypted with the KDC’s private key in steps 5 and 6. This binds the session key Ks to the identities of the two parties that will be engaged in the session. This inclusion of IDA accounts for the fact that the nonce value Na is considered unique only among all nonces generated by A, not among all nonces generated by all parties. Thus, it is the pair {IDA, Na}that uniquely identifies the connection request of A.

In both this example and the protocols described earlier, protocols that ap-peared secure were revised after additional analysis. These examples highlight the difficulty of getting things right in the area of authentication.


We have already presented public-key encryption approaches that are suited to electronic mail, including the straightforward encryption of the entire message for confidentiality (Figure 12.1b), authentication (Figure 12.1c), or both (Figure 12.1d). These approaches require that either the sender know the recipient’s public key (confidentiality), the recipient know the sender’s public key (authentication), or both (confidentiality plus authentication). In addition, the public-key algorithm must be applied once or twice to what may be a long message.

If confidentiality is the primary concern, then the following may be more efficient:

A S B: E(PUb, Ks) }E(Ks, M)

In this case, the message is encrypted with a one-time secret key. A also encrypts this one-time key with B’s public key. Only B will be able to use the corresponding private key to recover the one-time key and then use that key to decrypt the mes-sage. This scheme is more efficient than simply encrypting the entire message with B’s public key.


If authentication is the primary concern, then a digital signature may suffice, as was illustrated in Figure 13.2:

A S B: M } E(PRa, H(M))

This method guarantees that A cannot later deny having sent the message. However, this technique is open to another kind of fraud. Bob composes a mes-sage to his boss Alice that contains an idea that will save the company money. He appends his digital signature and sends it into the e-mail system. Eventually, the message will get delivered to Alice’s mailbox. But suppose that Max has heard of Bob’s idea and gains access to the mail queue before delivery. He finds Bob’s mes-sage, strips off his signature, appends his, and requeues the message to be delivered to Alice. Max gets credit for Bob’s idea.

To counter such a scheme, both the message and signature can be encrypted with the recipient’s public key:

A S B: E(PUb, [M }E(PRa, H(M))])

The latter two schemes require that B know A’s public key and be convinced that it is timely. An effective way to provide this assurance is the digital certificate, described in Chapter 14. Now we have

A S B: M }E(PRa, H(M)) }E(PRas, [T } IDA }PUa])

In addition to the message, A sends B the signature encrypted with A’s private key and A’s certificate encrypted with the private key of the authentication server. The recipient of the message first uses the certificate to obtain the sender’s public key and verify that it is authentic and then uses the public key to verify the message itself. If confidentiality is required, then the entire message can be encrypted with B’s public key. Alternatively, the entire message can be encrypted with a one-time secret key; the secret key is also transmitted, encrypted with B’s public key. This approach is explored in Chapter 19.

15.5 Federated identity management

Federated identity management is a relatively new concept dealing with the use of a common identity management scheme across multiple enterprises and numerous applications and supporting many thousands, even millions, of users. We begin our overview with a discussion of the concept of identity management and then examine federated identity management.

Identity Management

Identity management is a centralized, automated approach to provide enterprise-wide access to resources by employees and other authorized individuals. The focus of iden-tity management is defining an identity for each user (human or process), associating attributes with the identity, and enforcing a means by which a user can verify identity. The central concept of an identity management system is the use of single sign-on (SSO). SSO enables a user to access all network resources after a single authentication.

15.5 / FedeRAted identity mAnAgement 479

Typical services provided by a federated identity management system include the following:

• Point of contact: Includes authentication that a user corresponds to the user name provided, and management of user/server sessions.

• SSO protocol services: Provides a vendor-neutral security token service for supporting a single sign on to federated services.

• Trust services: Federation relationships require a trust relationship-based federation between business partners. A trust relationship is represented by the combination of the security tokens used to exchange information about a user, the cryptographic information used to protect these security tokens, and optionally the identity mapping rules applied to the information contained within this token.

• Key services: Management of keys and certificates.

• Identity services: services that provide the interface to local data stores, including user registries and databases, for identity-related information management.

• Authorization: Granting access to specific services and/or resources based on the authentication.

• Provisioning: Includes creating an account in each target system for the user, enrollment or registration of user in accounts, establishment of access rights or credentials to ensure the privacy and integrity of account data.

• Management: Services related to runtime configuration and deployment.

Note that Kerberos contains a number of the elements of an identity manage-ment system.

Figure 15.4 illustrates entities and data flows in a generic identity manage-ment architecture. A principal is an identity holder. Typically, this is a human user that seeks access to resources and services on the network. User devices, agent pro-cesses, and server systems may also function as principals. Principals authenticate themselves to an identity provider. The identity provider associates authentication information with a principal, as well as attributes and one or more identifiers.

Increasingly, digital identities incorporate attributes other than simply an identifier and authentication information (such as passwords and biometric information). An attribute service manages the creation and maintenance of such attributes. For example, a user needs to provide a shipping address each time an order is placed at a new Web merchant, and this information needs to be revised when the user moves. Identity management enables the user to provide this infor-mation once, so that it is maintained in a single place and released to data consum-ers in accordance with authorization and privacy policies. Users may create some of the attributes to be associated with their digital identity, such as an address. Administrators may also assign attributes to users, such as roles, access permis-sions, and employee information.

Data consumers are entities that obtain and employ data maintained and provided by identity and attribute providers, which are often used to support autho-rization decisions and to collect audit information. For example, a database server


or file server is a data consumer that needs a client’s credentials so as to know what access to provide to that client.

Identity Federation

Identity federation is, in essence, an extension of identity management to multiple security domains. Such domains include autonomous internal business units, external business partners, and other third-party applications and services. The goal is to provide the sharing of digital identities so that a user can be authen-ticated a single time and then access applications and resources across multiple domains. Because these domains are relatively autonomous or independent, no centralized control is possible. Rather, the cooperating organizations must form a federation based on agreed standards and mutual levels of trust to securely share digital identities.

Federated identity management refers to the agreements, standards, and technologies that enable the portability of identities, identity attributes, and entitle-ments across multiple enterprises and numerous applications and supporting many thousands, even millions, of users. When multiple organizations implement interop-erable federated identity schemes, an employee in one organization can use a single sign-on to access services across the federation with trust relationships associated with the identity. For example, an employee may log onto her corporate intranet and be authenticated to perform authorized functions and access authorized ser-vices on that intranet. The employee could then access their health benefits from an outside health-care provider without having to reauthenticate.

Beyond SSO, federated identity management provides other capabilities. One is a standardized means of representing attributes. Increasingly, digital identities incorporate attributes other than simply an identifier and authentication informa-tion (such as passwords and biometric information). Examples of attributes include account numbers, organizational roles, physical location, and file ownership. A user

IdentityProvider

AttributeService

Dataconsumer

Principal

Administrator

Figure 15.4 Generic Identity Management Architecture


may have multiple identifiers; for example, each identifier may be associated with a unique role with its own access permissions.

Another key function of federated identity management is identity mapping. Different security domains may represent identities and attributes differently. Further, the amount of information associated with an individual in one domain may be more than is necessary in another domain. The federated identity manage-ment protocols map identities and attributes of a user in one domain to the require-ments of another domain.

Figure 15.5 illustrates entities and data flows in a generic federated identity management architecture.

The identity provider acquires attribute information through dialogue and protocol exchanges with users and administrators. For example, a user needs to provide a shipping address each time an order is placed at a new Web merchant, and this information needs to be revised when the user moves. Identity manage-ment enables the user to provide this information once, so that it is maintained in a single place and released to data consumers in accordance with authorization and privacy policies.

User

1

Identity Provider(source domain)

Service Provider(destination domain)

1 End user’s browser or other application engagesin an authentication dialogue with identity providerin the same domain. End user also provides attributevalues associated with user’s identity.

2 Some attributes associated with an identity, such asallowable roles, may be provided by an administratorin the same domain.

3 A service provider in a remote domain, which the userwishes to access, obtains identity information, authentication information, and associated attributesfrom the identity provider in the source domain.

4 Service provider opens session with remote user andenforces access control restrictions based on user’sidentity and attributes.

Administrator

2

3

4

Figure 15.5 Federated Identity Operation


Service providers are entities that obtain and employ data maintained and pro-vided by identity providers, often to support authorization decisions and to collect audit information. For example, a database server or file server is a data consumer that needs a client’s credentials so as to know what access to provide to that client. A service provider can be in the same domain as the user and the identity provider. The power of this approach is for federated identity management, in which the service provider is in a different domain (e.g., a vendor or supplier network).

StAnDArDS Federated identity management uses a number of standards as the building blocks for secure identity exchange across different domains or heteroge-neous systems. In essence, organizations issue some form of security tickets for their users that can be processed by cooperating partners. Identity federation standards are thus concerned with defining these tickets, in terms of content and format, pro-viding protocols for exchanging tickets and performing a number of management tasks. These tasks include configuring systems to perform attribute transfers and identity mapping, and performing logging and auditing functions. The key standards are as follows:

• The Extensible Markup Language (XML): A markup language that uses sets of embedded tags or labels to characterize text elements within a document so as to indicate their appearance, function, meaning, or context. XML docu-ments appear similar to HTML (Hypertext Markup Language) documents that are visible as Web pages, but provide greater functionality. XML includes strict definitions of the data type of each field, thus supporting database for-mats and semantics. XML provides encoding rules for commands that are used to transfer and update data objects.

• The Simple Object Access Protocol (SOAP): A minimal set of conventions for invoking code using XML over HTTP. It enables applications to request services from one another with XML-based requests and receive responses as data for-matted with XML. Thus, XML defines data objects and structures, and SOAP provides a means of exchanging such data objects and performing remote proce-dure calls related to these objects. See [ROS06] for an informative discussion.

• WS-Security: A set of SOAP extensions for implementing message integrity and confidentiality in Web services. To provide for secure exchange of SOAP messages among applications, WS-Security assigns security tokens to each message for use in authentication.

• Security Assertion Markup Language (SAML): An XML-based language for the exchange of security information between online business partners. SAML conveys authentication information in the form of assertions about subjects. Assertions are statements about the subject issued by an authorita-tive entity.

The challenge with federated identity management is to integrate multiple technologies, standards, and services to provide a secure, user-friendly utility. The key, as in most areas of security and networking, is the reliance on a few mature standards widely accepted by industry. Federated identity management seems to have reached this level of maturity.


exAmpleS To get some feel for the functionality of identity federation, we look at three scenarios, taken from [COMP06].

In the first scenario (Figure 15.6a), Workplace.com contracts with Health.com to provide employee health benefits. An employee uses a Web interface to sign on to Workplace.com and goes through an authentication procedure there. This enables the employee to access authorized services and resources at Workplace.com. When the employee clicks on a link to access health benefits, her browser is redirected to Health.com. At the same time, the Workplace.com software passes the user’s identi-fier to Health.com in a secure manner. The two organizations are part of a federation that cooperatively exchanges user identifiers. Health.com maintains user identities for every employee at Workplace.com and associates with each identity health-benefits information and access rights. In this example, the linkage between the two companies is based on account information and user participation is browser based.

Figure 15.6b shows a second type of browser-based scheme. PartsSupplier.com is a regular supplier of parts to Workplace.com. In this case, a role-based access-control (RBAC) scheme is used for access to information. An engineer

User store

(a) Federation based on account linking

(c) Chained Web Services

Workplace.com(employee portal)

NameJoeJaneRavi

ID121314101603

User storeNameJoeJaneRavi

ID121314101603

Health.com

User store

(b) Federation based on roles

NameJoeJaneRavi

ID121314101603

DeptEngPurchPurch

User storeRole

EngineerPurchaser

Authentication Website access

End user(employee)

User ID

Workplace.com(procurementapplication)

PinSupplies.com(Purchasing Web

service)

Authentication

Procurement

request

End user(employee)

SOAP message

Eship.com(shipping Web

service)

SOAP message

Workplace.com(employee portal)

PartsSupplier.com

Authentication Website access

End user(employee)

Role

Figure 15.6 Federated Identity Scenarios


of Workplace.com authenticates at the employee portal at Workplace.com and clicks on a link to access information at PartsSupplier.com. Because the user is authenticated in the role of an engineer, he is taken to the technical doc-umentation and troubleshooting portion of PartsSupplier.com’s Web site with-out having to sign on. Similarly, an employee in a purchasing role signs on at Workplace.com and is authorized, in that role, to place purchases at PartsSupplier.com without having to authenticate to PartsSupplier.com. For this scenario, PartsSupplier.com does not have identity information for individual employees at Workplace.com. Rather, the linkage between the two federated partners is in terms of roles.

The scenario illustrated in Figure 15.6c can be referred to as document based rather than browser based. In this third example, Workplace.com has a purchasing agreement with PinSupplies.com, and PinSupplies.com has a business relationship with E-Ship.com. An employee of WorkPlace.com signs on and is authenticated to make purchases. The employee goes to a procure-ment application that provides a list of WorkPlace.com’s suppliers and the parts that can be ordered. The user clicks on the PinSupplies button and is presented with a purchase order Web page (HTML page). The employee fills out the form and clicks the submit button. The procurement application generates an XML/SOAP document that it inserts into the envelope body of an XML-based message. The procurement application then inserts the user’s credentials in the envelope header of the message, together with Workplace.com’s organizational identity. The procurement application posts the message to the PinSupplies.com’s purchasing Web service. This service authenticates the incoming message and processes the request. The purchasing Web service then sends a SOAP message to its shipping partner to fulfill the order. The message includes a PinSupplies.com security token in the envelope header and the list of items to be shipped as well as the end user’s shipping information in the envelope body. The shipping Web service authenticates the request and processes the shipment order.

15.6 PersOnaL identity veriFicatiOn

User authentication based on the possession of a smart card is becoming more wide-spread. A smart card has the appearance of a credit card, has an electronic inter-face, and may use a variety of authentication protocols.

A smart card contains within it an entire microprocessor, including processor, memory, and I/O ports. Some versions incorporate a special co-processing circuit for cryptographic operation to speed the task of encoding and decoding messages or generating digital signatures to validate the information transferred. In some cards, the I/O ports are directly accessible by a compatible reader by means of exposed electrical contacts. Other cards rely instead on an embedded antenna for wireless communication with the reader.

A typical smart card includes three types of memory. Read-only memory (ROM) stores data that does not change during the card’s life, such as the card

15.6 / PeRsonAl identity VeRiFicAtion 485

number and the cardholder’s name. Electrically erasable programmable ROM (EEPROM) holds application data and programs, such as the protocols that the card can execute. It also holds data that may vary with time. For example, in a tele-phone card, the EEPROM holds the talk time remaining. Random access memory (RAM) holds temporary data generated when applications are executed.

For the practical application of smart card authentication, a wide range of vendors must conform to standards that cover smart card protocols, authentication and access control formats and protocols, database entries, message formats, and so on. An important step in this direction is FIPS 201-2 (Personal Identity Verification [PIV] of Federal Employees and Contractors, June 2012). The standard defines a reliable, government-wide PIV system for use in applications such as access to fed-erally controlled facilities and information systems. The standard specifies a PIV system within which common identification credentials can be created and later used to verify a claimed identity. The standard also identifies Federal government-wide requirements for security levels that are dependent on risks to the facility or information being protected. The standard applies to private-sector contractors as well, and serves as a useful guideline for any organization.

PIV System Model

Figure 15.7 illustrates the major components of FIPS 201-2 compliant systems. The PIV front end defines the physical interface to a user who is requesting access to a facility, which could be either physical access to a protected physical area or logi-cal access to an information system. The PIV front-end subsystem supports up to three-factor authentication; the number of factors used depends on the level of se-curity required. The front end makes use of a smart card, known as a PIV card, which is a dual-interface contact and contactless card. The card holds a cardholder photograph, X.509 certificates, cryptographic keys, biometric data, and a cardholder unique identifier (CHUID). Certain cardholder information may be read-protected and require a personal identification number (PIN) for read access by the card reader. The biometric reader, in the current version of the standard, is a fingerprint reader or an iris scanner.

The standard defines three assurance levels for verification of the card and the encoded data stored on the card, which in turn leads to verifying the authenticity of the person holding the credential. A level of some confidence corresponds to use of the card reader and PIN. A level of high confidence adds a biometric comparison of a fingerprint captured and encoded on the card during the card-issuing process and a fingerprint scanned at the physical access point. A very high confidence level requires that the process just described is completed at a control point attended by an official observer.

The other major component of the PIV system is the PIV card issuance and management subsystem. This subsystem includes the components responsible for identity proofing and registration, card and key issuance and management, and the various repositories and services (e.g., public key infrastructure [PKI] directory, cer-tificate status servers) required as part of the verification infrastructure.

The PIV system interacts with a relying subsystem, which includes compo-nents responsible for determining a particular PIV cardholder’s access to a physical


Identity profiling& registration

Card issuance& maintenance

Keymanagement

PKI directory &certificate status

responder

Authorizationdata

Authorizationdata

Physicalresource

Logicalresource

PIV Card Issuanceand Management

Shapes

Relying

I&A Authorization

Physical Access Control

I&A = Identification and Authentication

Authorization

Direction of information flow

Processes

Components

Logical Access Control

Card reader/writer

PIN inputdevice

Biometricreader

PIV card

PIV Front End

LEGEND

I&A

Figure 15.7 FIPS 201 PIV System Model

or logical resource. FIPS 201-2 standardizes data formats and protocols for interac-tion between the PIV system and the relying system.

Unlike the typical card number/facility code encoded on most access control cards, the FIPS 201 CHUID takes authentication to a new level, through the use of an expiration date (a required CHUID data field) and an optional CHUID digital sig-nature. A digital signature can be checked to ensure that the CHUID recorded on the card was digitally signed by a trusted source and that the CHUID data have not been altered since the card was signed. The CHUID expiration date can be checked to verify that the card has not expired. This is independent from whatever expiration date is as-sociated with cardholder privileges. Reading and verifying the CHUID alone provides only some assurance of identity because it authenticates the card data, not the card-holder. The PIN and biometric factors provide identity verification of the individual.

PIV Documentation

The PIV specification is quite complex, and NIST has issued a number of docu-ments that cover a broad range of PIV topics. These are as follows:

• FIPS 201-2—Personal Identity Verification (PIV) of Federal Employees and Contractors: Specifies the physical card characteristics, storage media, and data elements that make up the identity credentials resident on the PIV card.


• SP 800-73-3—Interfaces for Personal Identity Verification: Specifies the inter-faces and card architecture for storing and retrieving identity credentials from a smart card, and provides guidelines for the use of authentication mecha-nisms and protocols.

• SP 800-76-2—Biometric Data Specification for Personal Identity Verification: Describes technical acquisition and formatting specifications for the biometric credentials of the PIV system.

• SP 800-78-3—Cryptographic Algorithms and Key Sizes for Personal Identity Verification: Identifies acceptable symmetric and asymmetric encryption algorithms, digital signature algorithms, and message digest algorithms, and specifies mechanisms to identify the algorithms associated with PIV keys or digital signatures.

• SP 800-104—A Scheme for PIV Visual Card Topography: Provides additional recommendations on the PIV card color-coding for designating employee affiliation.

• SP 800-116—A Recommendation for the Use of PIV Credentials in Physical Access Control Systems (PACS): Describes a risk-based approach for select-ing appropriate PIV authentication mechanisms to manage physical access to Federal government facilities and assets.

• SP 800-79-1—Guidelines for the Accreditation of Personal Identity Verification Card Issuers: Provides guidelines for accrediting the reliability of issuers of PIV cards that collect, store, and disseminate personal identity credentials and issue smart cards.

• SP 800-96—PIV Card to Reader Interoperability Guidelines: Provides requirements that facilitate interoperability between any card and any reader.

In addition there are other documents that deal with conformance testing and codes for identifiers.

PIV Credentials and Keys

The PIV card contains a number of mandatory and optional data elements that serve as identity credentials with varying levels of strength and assurance. These credentials are used singly or in sets to authenticate the holder of the PIV card to achieve the level of assurance required for a particular activity or transaction. The mandatory data elements are the following:

• Personal Identification Number (PIN): Required to activate the card for priv-ileged operation.

• Cardholder Unique Identifier (CHUID): Includes the Federal Agency Smart Credential Number (FASC-N) and the Global Unique Identification Number (GUID), which uniquely identify the card and the cardholder.

• PIV Authentication Key: Asymmetric key pair and corresponding certificate for user authentication.

• Two fingerprint templates: For biometric authentication.


• Electronic facial image: For biometric authentication.

• Asymmetric Card Authentication Key: Asymmetric key pair and correspond-ing certificate used for card authentication.

Optional elements include the following:

• Digital Signature Key: Asymmetric key pair and corresponding certificate that supports document signing and signing of data elements such as the CHUID.

• Key Management Key: Asymmetric key pair and corresponding certificate supporting key establishment and transport.

• Symmetric Card Authentication Key: For supporting physical access applications.

• PIV Card Application Administration Key: Symmetric key associated with the card management system.

• One or two iris images: For biometric authentication.

Table 15.5 lists the algorithm and key size requirements for PIV key types.

Authentication

Using the electronic credentials resident on a PIV card, the card supports the following authentication mechanisms:

• CHUID: The cardholder is authenticated using the signed CHUID data element on the card. The PIN is not required. This mechanism is useful in environments where a low level of assurance is acceptable and rapid contact-less authentication is necessary.

Table 15.5 PIV Algorithms and Key Sizes

PIV Key Type Algorithms Key Sizes (bits) Application

PIV Authentication Key

RSA 2048 Supports card and cardholder authentication for an interoperable environmentECDSA 256

Card Authentication Key

3TDEA 168 Supports card authentication for physical accessAES 128, 192, or 256

RSA 2048 Supports card authentication for an interoperable environmentECDSA 256

Digital Signature KeyRSA 2048 or 3072 Supports document signing

and nonce signingECDSA 256 or 384

Key Management KeyRSA 2048 Supports key establishment

and transportECDH 256 or 384


• Card Authentication Key: The PIV card is authenticated using the Card Authentication Key in a challenge response protocol. The PIN is not required. This mechanism allows contact (via card reader) or contactless (via radio waves) authentication of the PIV card without the holder’s active participa-tion, and provides a low level of assurance.

• BIO: The cardholder is authenticated by matching his or her fingerprint sample(s) to the signed biometric data element in an environment without a human attendant in view. The PIN is required to activate the card. This mech-anism achieves a high level of assurance and requires the cardholder’s active participation is submitting the PIN as well as the biometric sample.

• BIO-A: The cardholder is authenticated by matching his or her fingerprint sample(s) to the signed biometric data element in an environment with a human attendant in view. The PIN is required to activate the card. This mech-anism achieves a very high level of assurance when coupled with full trust validation of the biometric template retrieved from the card, and requires the cardholder’s active participation is submitting the PIN as well as the biomet-ric sample.

• PKI: The cardholder is authenticated by demonstrating control of the PIV authentication private key in a challenge response protocol that can be vali-dated using the PIV authentication certificate. The PIN is required to activate the card. This mechanism achieves a very high level of identity assurance and requires the cardholder’s knowledge of the PIN.

In each of the above use cases, except the symmetric Card Authentication Key use case, the source and the integrity of the corresponding PIV credential are vali-dated by verifying the digital signature on the credential, with the signature being provided by a trusted entity.

A variety of protocols can be constructed for each of these authentication types. SP-800-78-3 gives examples for each type. Figure 15.8 illustrates an authen-tication scenario that includes the use of the PIV Authentication Key. This scenario provides a high level of assurance. This scenario would be appropriate for authenti-cation of a user who possesses a PIV card and seeks access to a computer resource. The computer, designated local system in the figure, includes PIV application software and communicates to the card via an application program interface that enables the use of relatively high-level procedure calls. These high-level commands are converted into PIV commands that are issued to the card through a physical interface via a card reader or via a wireless interface. In either case, SP-800-73 refers to the card command interface as the PIV card edge.

The process begins when the local system detects the card either through an attached card reader or wirelessly. It then selects an application on the card for authentication. The local system then requests the public-key certificate for the card’s PIV Authentication Key. If the certificate is valid (i.e., has a valid sig-nature, has not expired or been revoked), authentication continues. Otherwise the card is rejected. The next step is for the local system to request that the card-holder enter the PIN for the card. If the submitted PIN matches the PIN stored on the card, the card returns a positive acknowledgment; otherwise the card returns


a failure message. The local system either continues or rejects the card accord-ingly. The next phase is a challenge-response protocol. The local system sends a nonce to be signed by the PIV, and the PIV returns the signature. The local system uses the PIV authentication public key to verify the signature. If the sig-nature is valid, the cardholder is accepted as being identified. Otherwise the local system rejects the card.

The scenario of Figure 15.8 accomplishes three types of authentication. The combination of possession of the card and knowledge of the PIN service authenti-cates the cardholder. The PIV Authentication Key certificate validates the card’s credentials. The challenge-response protocol authenticates the card.

Connect

Disconnect

End transaction

Verify PIN

Request card signature

PIV card app ID and Version

PIV Auth certificate returned

Signed nonce returned

PIN ACK

Read value (PIV Auth certificate)

Select application

Select application

Verify PIN

Sign nonce

Begin transaction

Present card(HolderV)

PIV Applicationon Local System

API onLocal System

PIVCard Edge

Retrieve PIVAUTH certificate

Retrieve FASC-Nfrom the certificate

CardV = Card validationCredV = Credential validationHolderV = Cardholder validationFASC-N = Federal Agency Smart Credential Number

Validate certificate(signature, expiration, and

revocation) (CredV)

Retrieve algorithmID and key size forsignature request

Acquire PIN(HolderV)

Verify signeddata—card

possesses privatekey (CardV)

Reject Cardholder identifier

Figure 15.8 Authentication Using PIV Authentication Key

15.8 / Key teRms, ReView Questions, And PRoblems 491


A painless way to get a grasp of Kerberos concepts is found in [BRYA88]. One of the best treatments of Kerberos is [KOHL94]. [TUNG99] describes Kerberos from a user’s point of view.

[SHIM05] provides a brief overview of federated identity management and examines one approach to standardization. [BHAT07] describes an integrated approach to federated identity management coupled with management of access control privileges.

BHAT07 Bhatti, R.; Bertino, E.; and Ghafoor, A. “An Integrated Approach to Federated Identity and Privilege Management in Open Systems.” Communications of the ACM, February 2007.

BRYA88 Bryant, W. Designing an Authentication System: A Dialogue in Four Scenes. Project Athena document, February 1988. Available at http://web.mit.edu/kerberos/www/dialogue.html.

KOHL94 Kohl, J.; Neuman, B.; and Ts’o, T. “The Evolution of the Kerberos Authentication Service.” in Brazier, F., and Johansen, D. Distributed Open Systems. Los Alamitos, CA: IEEE Computer Society Press, 1994. Available at http://web.mit .edu/kerberos/www/papers.html.

SHIM05 Shim, S.; Bhalla, G.; and Pendyala, V. “Federated Identity Management.” Computer, December 2005.

TUNG99 Tung, B. Kerberos: A Network Authentication System. Reading, MA: Addison-Wesley, 1999.


Key Terms

authenticationauthentication serverfederated identity

managementidentity managementKerberos

Kerberos realm mutual authenticationnonceone-way authenticationpersonal identity verification

(PIV)

realm replay attacksuppress-replay attackticketticket-granting server (TGS)timestamp

Review Questions 15.1 Give examples of replay attacks. 15.2 List three general approaches to dealing with replay attacks. 15.3 What is a suppress-replay attack? 15.4 What problem was Kerberos designed to address? 15.5 What are three threats associated with user authentication over a network or

Internet?

http://web.mit.edu/kerberos/www/dialogue.html


http://web.mit.edu/kerberos/www/papers.html



15.6 List three approaches to secure user authentication in a distributed environment. 15.7 What four requirements were defined for Kerberos? 15.8 What entities constitute a full-service Kerberos environment? 15.9 In the context of Kerberos, what is a realm? 15.10 What are the principal differences between version 4 and version 5 of Kerberos?

Problems 15.1 In Section 15.4, we outlined the public-key scheme proposed in [WOO92a] for the

distribution of secret keys. The revised version includes IDA in steps 5 and 6. What attack, specifically, is countered by this revision?

15.2 The protocol referred to in Problem 15.1 can be reduced from seven steps to five, hav-ing the following sequence:

a. A S B: b. A S KDC: c. KDC S B: d. B S A: e. A S B:

Show the message transmitted at each step. Hint: The final message in this protocol is the same as the final message in the original protocol.

15.3 Reference the suppress-replay attack described in Section 15.2 to answer the following.

a. Give an example of an attack when a party’s clock is ahead of that of the KDC. b. Give an example of an attack when a party’s clock is ahead of that of another

party. 15.4 There are three typical ways to use nonces as challenges. Suppose Na is a nonce

generated by A, A and B share key K, and f() is a function (such as an increment). The three usages are

Usage 1 Usage 2 Usage 3

(1) A S B: Na

(2) B S A: E(K, Na)(1) A S B: E(K, Na)(2) B S A: Na

(1) A S B: E(K, Na)(2) B S A: E(K, f(Na))

Describe situations for which each usage is appropriate. 15.5 Show that a random error in one block of ciphertext is propagated to all subsequent

blocks of plaintext in PCBC mode (See Figure T.2 in Appendix T). 15.6 Suppose that, in PCBC mode, blocks Ci and Ci+ 1 are interchanged during transmis-

sion. Show that this affects only the decrypted blocks Pi and Pi+ 1 but not subse-quent blocks.

15.7 In addition to providing a standard for public-key certificate formats, X.509 specifies an authentication protocol. The original version of X.509 contains a security flaw. The essence of the protocol is as follows.

A S B: A {tA, rA, IDB}

B S A: B {tB, rB, IDA, rA}

A S B: A {rB}

where tA and tB are timestamps, rA and rB are nonces and the notation X {Y} indicates that the message Y is transmitted, encrypted, and signed by X.

15.8 / Key teRms, ReView Questions, And PRoblems 493

The text of X.509 states that checking timestamps tA and tB is optional for three-way authentication. But consider the following example: Suppose A and B have used the preceding protocol on some previous occasion, and that opponent C has intercepted the preceding three messages. In addition, suppose that timestamps are not used and are all set to 0. Finally, suppose C wishes to impersonate A to B. C initially sends the first captured message to B:

C S B: A {0, rA, IDB}

B responds, thinking it is talking to A but is actually talking to C:

B S C: B {0, r′B, IDA, rA}

C meanwhile causes A to initiate authentication with C by some means. As a result, A sends C the following:

A S C: A {0, r′A, IDC}

C responds to A using the same nonce provided to C by B:

C S A: C {0, r′B, IDA, r′A}

A responds with

A S C: A {r′B}

This is exactly what C needs to convince B that it is talking to A, so C now repeats the incoming message back out to B.

C S B: A {r′B}

So B will believe it is talking to A whereas it is actually talking to C. Suggest a simple solution to this problem that does not involve the use of timestamps.

15.8 Consider a one-way authentication technique based on asymmetric encryption:

A S B: IDA

B S A: R1

A S B: E(PRa, R1)

a. Explain the protocol. b. What type of attack is this protocol susceptible to? 15.9 Consider a one-way authentication technique based on asymmetric encryption:

A S B: IDA

B S A: E(PUa, R2)

A S B: R2

a. Explain the protocol. b. What type of attack is this protocol susceptible to? 15.10 In Kerberos, when Bob receives a Ticket from Alice, how does he know it is genuine? 15.11 In Kerberos, when Bob receives a Ticket from Alice, how does he know it came from

Alice? 15.12 In Kerberos, when Alice receives a reply, how does she know it came from Bob (that

it’s not a replay of an earlier message from Bob)? 15.13 In Kerberos, what does the Ticket contain that allows Alice and Bob to talk securely?


495

16.1 Network Access Control

Elements of a Network Access Control SystemNetwork Access Enforcement Methods

16.2 Extensible Authentication Protocol

Authentication MethodsEAP Exchanges

16.3 IEEE 802.1X Port-Based Network Access Control

16.4 Cloud Computing

Cloud Computing ElementsCloud Computing Reference Architecture

16.5 Cloud Security Risks and Countermeasures

16.6 Data Protection in the Cloud

16.7 Cloud Security as a Service



Part 5: Network aNd INterNet SecurIty

Chapter

Network Access coNtrol ANd cloud security

495

496 Chapter 16 / Network aCCess CoNtrol aNd Cloud seCurity

Learning Objectives


u Discuss the principal elements of a network access control system.

u Discuss the principal network access enforcement methods.

u Present an overview of the Extensible Authentication Protocol.

u Understand the operation and role of the IEEE 802.1X Port-Based Network Access Control mechanism.

u Present an overview of cloud computing concepts.

u Understand the unique security issues related to cloud computing.

This chapter begins our discussion of network security, focusing on two key topics: network access control and cloud security. We begin with an overview of network access control systems, summarizing the principal elements and techniques involved in such a system. Next, we discuss the Extensible Authentication Protocol and IEEE 802.1X, two widely implemented standards that are the foundation of many network access control systems.

The remainder of the chapter deals with cloud security. We begin with an overview of cloud computing, and follow this with a discussion of cloud security issues.

16.1 netwOrk access cOntrOL

Network access control (NAC) is an umbrella term for managing access to a net-work. NAC authenticates users logging into the network and determines what data they can access and actions they can perform. NAC also examines the health of the user’s computer or mobile device (the endpoints).

Elements of a Network Access Control System

NAC systems deal with three categories of components:

• Access requestor (AR): The AR is the node that is attempting to access the network and may be any device that is managed by the NAC system, including workstations, servers, printers, cameras, and other IP-enabled devices. ARs are also referred to as supplicants, or simply, clients.

“No ticket! Dear me, Watson, this is really very singular. According to my experience it is not possible to reach the platform of a Metropolitan train without exhibiting one’s ticket.”

—The Adventure of the Bruce-Partington Plans, Sir Arthur Conan Doyle

16.1 / Network aCCess CoNtrol 497

• Policy server: Based on the AR’s posture and an enterprise’s defined policy, the policy server determines what access should be granted. The policy server often relies on backend systems, including antivirus, patch management, or a user directory, to help determine the host’s condition.

• Network access server (NAS): The NAS functions as an access control point for users in remote locations connecting to an enterprise’s internal network. Also called a media gateway, a remote access server (RAS), or a policy server, an NAS may include its own authentication services or rely on a separate authentication service from the policy server.

Figure 16.1 is a generic network access diagram. A variety of different ARs seek access to an enterprise network by applying to some type of NAS. The first step is generally to authenticate the AR. Authentication typically involves some sort of secure protocol and the use of cryptographic keys. Authentication may be performed by the NAS, or the NAS may mediate the authentication process. In the latter case, authentication takes place between the supplicant and an authentication server that is part of the policy server or that is accessed by the policy server.

Supplicants

Network access servers

Authenticationserver

DHCPserver

VLANserver

Policyserver

Patchmanagement

Networkresources

Quarantinenetwork

Antivirus Antispyware

Enterprise network

Figure 16.1 Network Access Control Context


The authentication process serves a number of purposes. It verifies a suppli-cant’s claimed identity, which enables the policy server to determine what access privileges, if any, the AR may have. The authentication exchange may result in the establishment of session keys to enable future secure communication between the supplicant and resources on the enterprise network.

Typically, the policy server or a supporting server will perform checks on the AR to determine if it should be permitted interactive remote access connectivity. These checks—sometimes called health, suitability, screening, or assessment checks—require software on the user’s system to verify compliance with certain re-quirements from the organization’s secure configuration baseline. For example, the user’s antimalware software must be up-to-date, the operating system must be fully patched, and the remote computer must be owned and controlled by the organiza-tion. These checks should be performed before granting the AR access to the enter-prise network. Based on the results of these checks, the organization can determine whether the remote computer should be permitted to use interactive remote access. If the user has acceptable authorization credentials but the remote computer does not pass the health check, the user and remote computer should be denied network access or have limited access to a quarantine network so that authorized personnel can fix the security deficiencies. Figure 16.1 indicates that the quarantine portion of the enterprise network consists of the policy server and related AR suitability serv-ers. There may also be application servers that do not require the normal security threshold be met.

Once an AR has been authenticated and cleared for a certain level of access to the enterprise network, the NAS can enable the AR to interact with resources in the enterprise network. The NAS may mediate every exchange to enforce a security policy for this AR, or may use other methods to limit the privileges of the AR.

Network Access Enforcement Methods

Enforcement methods are the actions that are applied to ARs to regulate access to the enterprise network. Many vendors support multiple enforcement methods simultaneously, allowing the customer to tailor the configuration by using one or a combination of methods. The following are common NAC enforcement methods.

• IEEE 802.1X: This is a link layer protocol that enforces authorization before a port is assigned an IP address. IEEE 802.1X makes use of the Extensible Authentication Protocol for the authentication process. Sections 16.2 and 16.3 cover the Extensible Authentication Protocol and IEEE 802.1X, respectively.

• Virtual local area networks (VLANs): In this approach, the enterprise net-work, consisting of an interconnected set of LANs, is segmented logically into a number of virtual LANs.1 The NAC system decides to which of the

1A VLAN is a logical subgroup within a LAN that is created via software rather than manually moving cables in the wiring closet. It combines user stations and network devices into a single unit regardless of the physical LAN segment they are attached to and allows traffic to flow more efficiently within popula-tions of mutual interest. VLANs are implemented in port-switching hubs and LAN switches.

16.2 / exteNsible autheNtiCatioN protoCol 499

network’s VLANs it will direct an AR, based on whether the device needs security remediation, Internet access only, or some level of network access to enterprise resources. VLANs can be created dynamically and VLAN mem-bership, of both enterprise servers and ARs, may overlap. That is, an enter-prise server or an AR may belong to more than one VLAN.

• Firewall: A firewall provides a form of NAC by allowing or denying network traffic between an enterprise host and an external user. Firewalls are discussed in Chapter 23.

• DHCP management: The Dynamic Host Configuration Protocol (DHCP) is an Internet protocol that enables dynamic allocation of IP addresses to hosts. A DHCP server intercepts DHCP requests and assigns IP addresses instead. Thus, NAC enforcement occurs at the IP layer based on subnet and IP assign-ment. A DCHP server is easy to install and configure, but is subject to various forms of IP spoofing, providing limited security.

There are a number of other enforcement methods available from vendors. The ones in the preceding list are perhaps the most common, and IEEE 802.1X is by far the most commonly implemented solution.

16.2 extensibLe authenticatiOn PrOtOcOL

The Extensible Authentication Protocol (EAP), defined in RFC 3748, acts as a framework for network access and authentication protocols. EAP provides a set of protocol messages that can encapsulate various authentication methods to be used between a client and an authentication server. EAP can operate over a vari-ety of network and link level facilities, including point-to-point links, LANs, and other networks, and can accommodate the authentication needs of the various links and networks. Figure 16.2 illustrates the protocol layers that form the con-text for EAP.

Authenticationmethods

EAPlayer

Data linklayer

Extensible Authentication Protocol (EAP)

IEEE 802.1XEAP over LAN (EAPOL)

EAP-TLS

EAP-TTLS

EAP-PSK

EAP-IKEv2

PPP802.3

Ethernet802.11WLAN

Other

Other

Figure 16.2 EAP Layered Context


Authentication Methods

EAP supports multiple authentication methods. This is what is meant by referring to EAP as extensible. EAP provides a generic transport service for the exchange of au-thentication information between a client system and an authentication server. The basic EAP transport service is extended by using a specific authentication protocol, or method, that is installed in both the EAP client and the authentication server.

Numerous methods have been defined to work over EAP. The following are commonly supported EAP methods:

• EAP-TLS (EAP Transport Layer Security): EAP-TLS (RFC 5216) defines how the TLS protocol (described in Chapter 17) can be encapsulated in EAP messages. EAP-TLS uses the handshake protocol in TLS, not its encryption method. Client and server authenticate each other using digital certificates. Client generates a pre-master secret key by encrypting a random number with the server’s public key and sends it to the server. Both client and server use the pre-master to generate the same secret key.

• EAP-TTLS (EAP Tunneled TLS): EAP-TTLS is like EAP-TLS, except only the server has a certificate to authenticate itself to the client first. As in EAP-TLS, a secure connection (the “tunnel”) is established with secret keys, but that connection is used to continue the authentication process by authenti-cating the client and possibly the server again using any EAP method or legacy method such as PAP (Password Authentication Protocol) and CHAP (Challenge-Handshake Authentication Protocol). EAP-TTLS is defined in RFC 5281.

• EAP-GPSK (EAP Generalized Pre-Shared Key): EAP-GPSK, defined in RFC 5433, is an EAP method for mutual authentication and session key deri-vation using a Pre-Shared Key (PSK). EAP-GPSK specifies an EAP method based on pre-shared keys and employs secret key-based cryptographic algo-rithms. Hence, this method is efficient in terms of message flows and com-putational costs, but requires the existence of pre-shared keys between each peer and EAP server. The set up of these pairwise secret keys is part of the peer registration, and thus, must satisfy the system preconditions. It provides a protected communication channel when mutual authentication is success-ful for both parties to communicate over and is designed for authentication over insecure networks such as IEEE 802.11. EAP-GPSK does not require any public-key cryptography. The EAP method protocol exchange is done in a minimum of four messages.

• EAP-IKEv2: It is based on the Internet Key Exchange protocol version 2 (IKEv2), which is described in Chapter 20. It supports mutual authentication and session key establishment using a variety of methods. EAP-TLS is defined in RFC 5106.

EAP Exchanges

Whatever method is used for authentication, the authentication information and authentication protocol information are carried in EAP messages.

16.2 / exteNsible autheNtiCatioN protoCol 501

RFC 3748 defines the goal of the exchange of EAP messages to be successful authentication. In the context of RFC 3748, successful authentication is an exchange of EAP messages, as a result of which the authenticator decides to allow access by the peer, and the peer decides to use this access. The authenticator’s decision typically involves both authentication and authorization aspects; the peer may successfully authenticate to the authenticator, but access may be denied by the authenticator due to policy reasons.

Figure 16.3 indicates a typical arrangement in which EAP is used. The follow-ing components are involved:

• EAP peer: Client computer that is attempting to access a network.

• EAP authenticator: An access point or NAS that requires EAP authentica-tion prior to granting access to a network.

• Authentication server: A server computer that negotiates the use of a spe-cific EAP method with an EAP peer, validates the EAP peer’s credentials, and authorizes access to the network. Typically, the authentication server is a Remote Authentication Dial-In User Service (RADIUS) server.

The authentication server functions as a backend server that can authenticate peers as a service to a number of EAP authenticators. The EAP authenticator then makes the decision of whether to grant access. This is referred to as the EAP pass-through mode. Less commonly, the authenticator takes over the role of the EAP server; that is, only two parties are involved in the EAP execution.

As a first step, a lower-level protocol, such as PPP (point-to-point protocol) or IEEE 802.1X, is used to connect to the EAP authenticator. The soft-ware entity in the EAP peer that operates at this level is referred to as the supplicant. EAP messages containing the appropriate information for a chosen EAP method are then exchanged between the EAP peer and the authentication server.

Method

EAP peer/Authenticator

EAP layer

Lower layer

EAPauthenticator

EAP layer

Lower layer

Method

EAP peer/Authenticator

EAP layer

Lower layerRADIUS

EAPmessages

EAPmessages

802.1X,PPP

EAP peerEAP authenticator Authentication server

(RADIUS)

Figure 16.3 EAP Protocol Exchanges


EAP messages may include the following fields:

• Code: Identifies the Type of EAP message. The codes are Request (1), Response (2), Success (3), and Failure (4).

• Identifier: Used to match Responses with Requests.

• Length: Indicates the length, in octets, of the EAP message, including the Code, Identifier, Length, and Data fields.

• Data: Contains information related to authentication. Typically, the Data field consists of a Type subfield, indicating the type of data carried, and a Type-Data field.

The Success and Failure messages do not include a Data field.The EAP authentication exchange proceeds as follows. After a lower-level

exchange that established the need for an EAP exchange, the authenticator sends a Request to the peer to request an identity, and the peer sends a Response with the identity information. This is followed by a sequence of Requests by the authentica-tor and Responses by the peer for the exchange of authentication information. The information exchanged and the number of Request–Response exchanges needed depend on the authentication method. The conversation continues until either (1) the authenticator determines that it cannot authenticate the peer and transmits an EAP Failure or (2) the authenticator determines that successful authentication has occurred and transmits an EAP Success.

Figure 16.4 gives an example of an EAP exchange. Not shown in the figure is a message or signal sent from the EAP peer to the authenticator using some protocol

EAP peer

EAP-Response/Identity

EAP-Request/Identity

EAP authenticator Authentication server(RADIUS)

EAP-Response/Auth

EAP-Request/Auth

EAP-Response/Auth

EAP-Request/Auth

EAP-Success/Failure

Figure 16.4 EAP Message Flow in Pass-Through Mode

16.3 / ieee 802.1x port-based Network aCCess CoNtrol 503

other than EAP and requesting an EAP exchange to grant network access. One protocol used for this purpose is IEEE 802.1X, discussed in the next section. The first pair of EAP Request and Response messages is of Type identity, in which the authenticator requests the peer’s identity, and the peer returns its claimed identity in the Response message. This Response is passed through the authenticator to the authentication server. Subsequent EAP messages are exchanged between the peer and the authentication server.

Upon receiving the identity Response message from the peer, the server selects an EAP method and sends the first EAP message with a Type field related to an authentication method. If the peer supports and accepts the selected EAP method, it replies with the corresponding Response message of the same type. Otherwise, the peer sends a NAK, and the EAP server either selects another EAP method or aborts the EAP execution with a failure message. The selected EAP method determines the number of Request/Response pairs. During the exchange the appropriate authentication information, including key material, is exchanged. The exchange ends when the server determines that authentication has succeeded or that no further attempt can be made and authentication has failed.

16.3 ieee 802.1x POrt-based netwOrk access cOntrOL

IEEE 802.1X Port-Based Network Access Control was designed to provide access con-trol functions for LANs. Table 16.1 briefly defines key terms used in the IEEE 802.11 standard. The terms supplicant, network access point, and authentication server cor-respond to the EAP terms peer, authenticator, and authentication server, respectively.

Until the AS authenticates a supplicant (using an authentication protocol), the authenticator only passes control and authentication messages between the sup-plicant and the AS; the 802.1X control channel is unblocked, but the 802.11 data channel is blocked. Once a supplicant is authenticated and keys are provided, the authenticator can forward data from the supplicant, subject to predefined access control limitations for the supplicant to the network. Under these circumstances, the data channel is unblocked.

As indicated in Figure 16.5, 802.1X uses the concepts of controlled and uncon-trolled ports. Ports are logical entities defined within the authenticator and refer to physical network connections. Each logical port is mapped to one of these two types of physical ports. An uncontrolled port allows the exchange of protocol data units (PDUs) between the supplicant and the AS, regardless of the authentication state of the supplicant. A controlled port allows the exchange of PDUs between a sup-plicant and other systems on the network only if the current state of the supplicant authorizes such an exchange.

The essential element defined in 802.1X is a protocol known as EAPOL (EAP over LAN). EAPOL operates at the network layers and makes use of an IEEE 802 LAN, such as Ethernet or Wi-Fi, at the link level. EAPOL enables a supplicant to communicate with an authenticator and supports the exchange of EAP packets for authentication.

The most common EAPOL packets are listed in Table 16.2. When the supplicant first connects to the LAN, it does not know the MAC address of the


authenticator. Actually it doesn’t know whether there is an authenticator present at all. By sending an EAPOL-Start packet to a special group-multicast address reserved for IEEE 802.1X authenticators, a supplicant can determine whether an authenticator is present and let it know that the supplicant is ready. In many cases, the authenticator will already be notified that a new device has connected from some hardware notification. For example, a hub knows that a cable is plugged in before the device sends any data. In this case the authenticator may preempt the Start message with its own message. In either case the authenticator sends an EAP-Request Identity message encapsulated in an EAPOL-EAP packet. The EAPOL-EAP is the EAPOL frame type used for transporting EAP packets.

The authenticator uses the EAP-Key packet to send cryptographic keys to the supplicant once it has decided to admit it to the network. The EAP-Logoff packet type indicates that the supplicant wishes to be disconnected from the network.

The EAPOL packet format includes the following fields:

• Protocol version: version of EAPOL.

• Packet type: indicates start, EAP, key, logoff, etc.

Table 16.1 Terminology Related to IEEE 802.1X

AuthenticatorAn entity at one end of a point-to-point LAN segment that facilities authentication of the entity to the other end of the link.

Authentication exchangeThe two-party conversation between systems performing an authentication process.

Authentication processThe cryptographic operations and supporting data frames that perform the actual authentication.

Authentication server (AS)An entity that provides an authentication service to an authenticator. This service determines, from the cre-dentials provided by supplicant, whether the supplicant is authorized to access the services provided by the system in which the authenticator resides.

Authentication transportThe datagram session that actively transfers the authentication exchange between two systems.

Bridge portA port of an IEEE 802.1D or 802.1Q bridge.

Edge portA bridge port attached to a LAN that has no other bridges attached to it.

Network access portA point of attachment of a system to a LAN. It can be a physical port, such as a single LAN MAC attached to a physical LAN segment, or a logical port, for example, an IEEE 802.11 association between a station and an access point.

Port access entity (PAE)The protocol entity associated with a port. It can support the protocol functionality associated with the authenticator, the supplicant, or both.

SupplicantAn entity at one end of a point-to-point LAN segment that seeks to be authenticated by an authenticator attached to the other end of that link.

16.4 / Cloud ComputiNg 505

• Packet body length: If the packet includes a body, this field indicates the body length.

• Packet body: The payload for this EAPOL packet. An example is an EAP packet.

Figure 16.6 shows an example of exchange using EAPOL. In Chapter 18, we exam-ine the use of EAP and EAPOL in the context of IEEE 802.11 wireless LAN security.

16.4 cLOud cOmPuting

There is an increasingly prominent trend in many organizations to move a substantial portion of or even all information technology (IT) operations to an Internet-connected infrastructure known as enterprise cloud computing. This section provides an overview of cloud computing.

Supplicant

Networkaccess point

Uncontrolledport

Controlledport

Authentication server

Network or Internet

Figure 16.5 802.1X Access Control

Table 16.2 Common EAPOL Frame Types

Frame Type Definition

EAPOL-EAP Contains an encapsulated EAP packet.

EAPOL-Start A supplicant can issue this packet instead of waiting for a challenge from the authenticator.

EAPOL-Logoff Used to return the state of the port to unauthorized when the supplicant if finished using the network.

EAPOL-Key Used to exchange cryptographic keying information.


Cloud Computing Elements

NIST defines cloud computing, in NIST SP-800-145 (The NIST Definition of Cloud Computing), as follows:

EAP peer

EAPOL-Start

EAPOL-EAP (EAP-Request/Identity)

EAPOL-EAP (EAP-Response/Identity)

EAP authenticator Authentication server(RADIUS)

EAPOL-Logoff

EAPOL-EAP (EAP-Response/Auth)

EAPOL-EAP (EAP-Request/Auth)

EAPOL-EAP (EAP-Response/Auth)

EAPOL-EAP (EAP-Request/Auth)

EAPOL-EAP (EAP-Success)

Figure 16.6 Example Timing Diagram for IEEE 802.1X

Cloud computing: A model for enabling ubiquitous, convenient, on-demand net-work access to a shared pool of configurable computing resources (e.g., networks, servers, storage, applications, and services) that can be rapidly provisioned and released with minimal management effort or service provider interaction. This cloud model promotes availability and is composed of five essential characteris-tics, three service models, and four deployment models.

The definition refers to various models and characteristics, whose relationship is illustrated in Figure 16.7. The essential characteristics of cloud computing include the following:

• Broad network access: Capabilities are available over the network and ac-cessed through standard mechanisms that promote use by heterogeneous thin


or thick client platforms (e.g., mobile phones, laptops, and PDAs) as well as other traditional or cloud-based software services.

• Rapid elasticity: Cloud computing gives you the ability to expand and reduce resources according to your specific service requirement. For example, you may need a large number of server resources for the duration of a specific task. You can then release these resources upon completion of the task.

• Measured service: Cloud systems automatically control and optimize resource use by leveraging a metering capability at some level of abstraction appropri-ate to the type of service (e.g., storage, processing, bandwidth, and active user accounts). Resource usage can be monitored, controlled, and reported, pro-viding transparency for both the provider and consumer of the utilized service.

• On-demand self-service: A consumer can unilaterally provision computing capabilities, such as server time and network storage, as needed automati-cally without requiring human interaction with each service provider. Because the service is on demand, the resources are not permanent parts of your IT infrastructure.

• Resource pooling: The provider’s computing resources are pooled to serve multiple consumers using a multi-tenant model, with different physical and virtual resources dynamically assigned and reassigned according to consumer

BroadNetwork Access

Resource Pooling

RapidElasticity

Ess

enti

alC

hara

cter

isti

csSe

rvic

eM

odel

sD

eplo

ymen

tM

odel

s

MeasuredService

On-DemandSelf-Service

Public Private Hybrid Community

Software as a Service (SaaS)

Platform as a Service (PaaS)

Infrastructure as a Service (IaaS)

Figure 16.7 Cloud Computing Elements


demand. There is a degree of location independence in that the customer generally has no control or knowledge of the exact location of the provided resources, but may be able to specify location at a higher level of abstrac-tion (e.g., country, state, or data center). Examples of resources include stor-age, processing, memory, network bandwidth, and virtual machines. Even private clouds tend to pool resources between different parts of the same organization.

NIST defines three service models, which can be viewed as nested service alternatives:

• Software as a service (SaaS): The capability provided to the consumer is to use the provider’s applications running on a cloud infrastructure. The appli-cations are accessible from various client devices through a thin client inter-face such as a Web browser. Instead of obtaining desktop and server licenses for software products it uses, an enterprise obtains the same functions from the cloud service. SaaS saves the complexity of software installation, main-tenance, upgrades, and patches. Examples of services at this level are Gmail, Google’s e-mail service, and Salesforce.com, which helps firms keep track of their customers.

• Platform as a service (PaaS): The capability provided to the consumer is to deploy onto the cloud infrastructure consumer-created or acquired applica-tions created using programming languages and tools supported by the pro-vider. PaaS often provides middleware-style services such as database and component services for use by applications. In effect, PaaS is an operating system in the cloud.

• Infrastructure as a service (IaaS): The capability provided to the consumer is to provision processing, storage, networks, and other fundamental computing resources where the consumer is able to deploy and run arbitrary software, which can include operating systems and applications. IaaS enables customers to combine basic computing services, such as number crunching and data stor-age, to build highly adaptable computer systems.

NIST defines four deployment models:

• Public cloud: The cloud infrastructure is made available to the general public or a large industry group and is owned by an organization selling cloud ser-vices. The cloud provider is responsible both for the cloud infrastructure and for the control of data and operations within the cloud.

• Private cloud: The cloud infrastructure is operated solely for an organiza-tion. It may be managed by the organization or a third party and may exist on premise or off premise. The cloud provider (CP) is responsible only for the infrastructure and not for the control.

• Community cloud: The cloud infrastructure is shared by several organizations and supports a specific community that has shared concerns (e.g., mission, security requirements, policy, and compliance considerations). It may be man-aged by the organizations or a third party and may exist on premise or off premise.


• Hybrid cloud: The cloud infrastructure is a composition of two or more clouds (private, community, or public) that remain unique entities but are bound to-gether by standardized or proprietary technology that enables data and appli-cation portability (e.g., cloud bursting for load balancing between clouds).

Figure 16.8 illustrates the typical cloud service context. An enterprise main-tains workstations within an enterprise LAN or set of LANs, which are connected by a router through a network or the Internet to the cloud service provider. The cloud service provider maintains a massive collection of servers, which it manages with a variety of network management, redundancy, and security tools. In the fig-ure, the cloud infrastructure is shown as a collection of blade servers, which is a common architecture.

Networkor Internet

Router

Router

Servers

LANswitch

LANswitch

Enterprisecloud user

Cloudservice

provider

Figure 16.8 Cloud Computing Context


Cloud Computing Reference Architecture

NIST SP 500-292 (NIST Cloud Computing Reference Architecture) establishes a ref-erence architecture, described as follows:

CloudConsumer

CloudAuditor

ServiceIntermediation

ServiceAggregation

ServiceArbitrage

CloudBroker

Cloud Provider

SecurityAudit

PerformanceAudit

PrivacyImpact Audit

SaaS

Service Layer

Service Orchestration CloudService

Management

PaaS

Hardware

Physical Resource Layer

Facility

Resource Abstractionand Control Layer

IaaS

BusinessSupport

Provisioning/Configuration

Portability/Interoperability

Secu

rity

Pri

vacy

Cloud Carrier

Figure 16.9 NIST Cloud Computing Reference Architecture

The NIST cloud computing reference architecture focuses on the requirements of “what” cloud services provide, not a “how to” design solution and implemen-tation. The reference architecture is intended to facilitate the understanding of the operational intricacies in cloud computing. It does not represent the system architecture of a specific cloud computing system; instead it is a tool for describ-ing, discussing, and developing a system-specific architecture using a common framework of reference.

NIST developed the reference architecture with the following objectives in mind:

• to illustrate and understand the various cloud services in the context of an overall cloud computing conceptual model

• to provide a technical reference for consumers to understand, discuss, catego-rize, and compare cloud services

• to facilitate the analysis of candidate standards for security, interoperability, and portability and reference implementations

The reference architecture, depicted in Figure 16.9, defines five major actors in terms of the roles and responsibilities:

• Cloud consumer: A person or organization that maintains a business relation-ship with, and uses service from, cloud providers.


• Cloud provider: A person, organization, or entity responsible for making a service available to interested parties.

• Cloud auditor: A party that can conduct independent assessment of cloud ser-vices, information system operations, performance, and security of the cloud implementation.

• Cloud broker: An entity that manages the use, performance, and delivery of cloud services, and negotiates relationships between CPs and cloud consumers.

• Cloud carrier: An intermediary that provides connectivity and transport of cloud services from CPs to cloud consumers.

The roles of the cloud consumer and provider have already been discussed. To summarize, a cloud provider can provide one or more of the cloud services to meet IT and business requirements of cloud consumers. For each of the three service models (SaaS, PaaS, IaaS), the CP provides the storage and processing facilities needed to support that service model, together with a cloud interface for cloud ser-vice consumers. For SaaS, the CP deploys, configures, maintains, and updates the operation of the software applications on a cloud infrastructure so that the services are provisioned at the expected service levels to cloud consumers. The consumers of SaaS can be organizations that provide their members with access to software ap-plications, end users who directly use software applications, or software application administrators who configure applications for end users.

For PaaS, the CP manages the computing infrastructure for the platform and runs the cloud software that provides the components of the platform, such as run-time software execution stack, databases, and other middleware components. Cloud consumers of PaaS can employ the tools and execution resources provided by CPs to develop, test, deploy, and manage the applications hosted in a cloud environment.

For IaaS, the CP acquires the physical computing resources underlying the service, including the servers, networks, storage, and hosting infrastructure. The IaaS cloud consumer in turn uses these computing resources, such as a virtual com-puter, for their fundamental computing needs.

The cloud carrier is a networking facility that provides connectivity and trans-port of cloud services between cloud consumers and CPs. Typically, a CP will set up service level agreements (SLAs) with a cloud carrier to provide services consistent with the level of SLAs offered to cloud consumers, and may require the cloud car-rier to provide dedicated and secure connections between cloud consumers and CPs.

A cloud broker is useful when cloud services are too complex for a cloud con-sumer to easily manage. Three areas of support can be offered by a cloud broker:

• Service intermediation: These are value-added services, such as identity man-agement, performance reporting, and enhanced security.

• Service aggregation: The broker combines multiple cloud services to meet consumer needs not specifically addressed by a single CP, or to optimize per-formance or minimize cost.

• Service arbitrage: This is similar to service aggregation except that the services being aggregated are not fixed. Service arbitrage means a broker has the flexibil-ity to choose services from multiple agencies. The cloud broker, for example, can use a credit-scoring service to measure and select an agency with the best score.


A cloud auditor can evaluate the services provided by a CP in terms of secu-rity controls, privacy impact, performance, and so on. The auditor is an independent entity that can assure that the CP conforms to a set of standards.

16.5 cLOud security risks and cOuntermeasures

In general terms, security controls in cloud computing are similar to the security controls in any IT environment. However, because of the operational models and technologies used to enable cloud service, cloud computing may present risks that are specific to the cloud environment. The essential concept in this regard is that the enterprise loses a substantial amount of control over resources, services, and appli-cations but must maintain accountability for security and privacy policies.

The Cloud Security Alliance [CSA10] lists the following as the top cloud-specific security threats, together with suggested countermeasures:

• Abuse and nefarious use of cloud computing: For many CPs, it is relatively easy to register and begin using cloud services, some even offering free limited trial periods. This enables attackers to get inside the cloud to conduct various attacks, such as spamming, malicious code attacks, and denial of service. PaaS providers have traditionally suffered most from this kind of attacks; however, recent evidence shows that hackers have begun to target IaaS vendors as well. The burden is on the CP to protect against such attacks, but cloud service cli-ents must monitor activity with respect to their data and resources to detect any malicious behavior.

Countermeasures include (1) stricter initial registration and valida-tion processes; (2) enhanced credit card fraud monitoring and coordination; (3) comprehensive introspection of customer network traffic; and (4) monitor-ing public blacklists for one’s own network blocks.

• Insecure interfaces and APIs: CPs expose a set of software interfaces or APIs that customers use to manage and interact with cloud services. The security and availability of general cloud services are dependent upon the security of these basic APIs. From authentication and access control to encryption and activity monitoring, these interfaces must be designed to protect against both accidental and malicious attempts to circumvent policy.

Countermeasures include (1) analyzing the security model of CP interfaces; (2) ensuring that strong authentication and access controls are implemented in concert with encrypted transmission; and (3) understanding the dependency chain associated with the API.

• Malicious insiders: Under the cloud computing paradigm, an organization re-linquishes direct control over many aspects of security and, in doing so, con-fers an unprecedented level of trust onto the CP. One grave concern is the risk of malicious insider activity. Cloud architectures necessitate certain roles that are extremely high risk. Examples include CP system administrators and man-aged security service providers.

Countermeasures include the following: (1) enforce strict supply chain management and conduct a comprehensive supplier assessment; (2) specify

16.5 / Cloud seCurity risks aNd CouNtermeasures 513

human resource requirements as part of legal contract; (3) require transpar-ency into overall information security and management practices, as well as compliance reporting; and (4) determine security breach notification processes.

• Shared technology issues: IaaS vendors deliver their services in a scalable way by sharing infrastructure. Often, the underlying components that make up this infrastructure (CPU caches, GPUs, etc.) were not designed to offer strong isolation properties for a multi-tenant architecture. CPs typically approach this risk by the use of isolated virtual machines for individual clients. This ap-proach is still vulnerable to attack, by both insiders and outsiders, and so can only be a part of an overall security strategy.

Countermeasures include the following: (1) implement security best practices for installation/configuration; (2) monitor environment for unau-thorized changes/activity; (3) promote strong authentication and access con-trol for administrative access and operations; (4) enforce SLAs for patching and vulnerability remediation; and (5) conduct vulnerability scanning and configuration audits.

• Data loss or leakage: For many clients, the most devastating impact from a security breach is the loss or leakage of data. We address this issue in the next subsection.

Countermeasures include the following: (1) implement strong API ac-cess control; (2) encrypt and protect integrity of data in transit; (3) analyze data protection at both design and run time; and (4) implement strong key generation, storage and management, and destruction practices.

• Account or service hijacking: Account or service hijacking, usually with sto-len credentials, remains a top threat. With stolen credentials, attackers can often access critical areas of deployed cloud computing services, allowing them to compromise the confidentiality, integrity, and availability of those services.

Countermeasures include the following: (1) prohibit the sharing of account credentials between users and services; (2) leverage strong two-factor authentication techniques where possible; (3) employ proactive monitoring to detect unauthorized activity; and (4) understand CP security policies and SLAs.

• Unknown risk profile: In using cloud infrastructures, the client necessarily cedes control to the CP on a number of issues that may affect security. Thus the client must pay attention to and clearly define the roles and responsibili-ties involved for managing risks. For example, employees may deploy applica-tions and data resources at the CP without observing the normal policies and procedures for privacy, security, and oversight.

Countermeasures include (1) disclosure of applicable logs and data; (2) partial/full disclosure of infrastructure details (e.g., patch levels and firewalls); and (3) monitoring and alerting on necessary information.

Similar lists have been developed by the European Network and Information Security Agency [ENIS09] and NIST [JANS11].


16.6 data PrOtectiOn in the cLOud

As can be seen from the previous section, there are numerous aspects to cloud security and numerous approaches to providing cloud security measures. A further example is seen in the NIST guidelines for cloud security, specified in SP-800-14 and listed in Table 16.3. Thus, the topic of cloud security is well beyond the scope of this chapter. In this section, we focus on one specific element of cloud security.

Table 16.3 NIST Guidelines on Security and Privacy Issues and Recommendations

GovernanceExtend organizational practices pertaining to the policies, procedures, and standards used for application development and service provisioning in the cloud, as well as the design, implementation, testing, use, and monitoring of deployed or engaged services.

Put in place audit mechanisms and tools to ensure organizational practices are followed throughout the system life cycle.

ComplianceUnderstand the various types of laws and regulations that impose security and privacy obligations on the orga-nization and potentially impact cloud computing initiatives, particularly those involving data location, privacy and security controls, records management, and electronic discovery requirements.

Review and assess the cloud provider’s offerings with respect to the organizational requirements to be met and ensure that the contract terms adequately meet the requirements.

Ensure that the cloud provider’s electronic discovery capabilities and processes do not compromise the privacy or security of data and applications.

TrustEnsure that service arrangements have sufficient means to allow visibility into the security and privacy controls and processes employed by the cloud provider, and their performance over time.

Establish clear, exclusive ownership rights over data.Institute a risk management program that is flexible enough to adapt to the constantly evolving and shift-

ing risk landscape for the life cycle of the system.Continuously monitor the security state of the information system to support ongoing risk management

decisions.

ArchitectureUnderstand the underlying technologies that the cloud provider uses to provision services, including the impli-cations that the technical controls involved have on the security and privacy of the system, over the full system life cycle and across all system components.

Identity and access managementEnsure that adequate safeguards are in place to secure authentication, authorization, and other identity and access management functions, and are suitable for the organization.

Software isolationUnderstand virtualization and other logical isolation techniques that the cloud provider employs in its multi-tenant software architecture, and assess the risks involved for the organization.

Data protectionEvaluate the suitability of the cloud provider’s data management solutions for the organizational data concerned and the ability to control access to data, to secure data while at rest, in transit, and in use, and to sanitize data.

Take into consideration the risk of collating organizational data with those of other organizations whose threat profiles are high or whose data collectively represent significant concentrated value.

16.6 / data proteCtioN iN the Cloud 515

Fully understand and weigh the risks involved in cryptographic key management with the facilities avail-able in the cloud environment and the processes established by the cloud provider.

AvailabilityUnderstand the contract provisions and procedures for availability, data backup and recovery, and disaster recovery, and ensure that they meet the organization’s continuity and contingency planning requirements.

Ensure that during an intermediate or prolonged disruption or a serious disaster, critical operations can be immediately resumed, and that all operations can be eventually reinstituted in a timely and organized manner.

Incident responseUnderstand the contract provisions and procedures for incident response and ensure that they meet the requirements of the organization.

Ensure that the cloud provider has a transparent response process in place and sufficient mechanisms to share information during and after an incident.

Ensure that the organization can respond to incidents in a coordinated fashion with the cloud provider in accordance with their respective roles and responsibilities for the computing environment.

There are many ways to compromise data. Deletion or alteration of records without a backup of the original content is an obvious example. Unlinking a record from a larger context may render it unrecoverable, as can storage on unreliable media. Loss of an encoding key may result in effective destruction. Finally, unau-thorized parties must be prevented from gaining access to sensitive data.

The threat of data compromise increases in the cloud, due to the number of and interactions between risks and challenges that are either unique to the cloud or more dangerous because of the architectural or operational characteristics of the cloud environment.

Database environments used in cloud computing can vary significantly. Some providers support a multi-instance model, which provides a unique DBMS running on a virtual machine instance for each cloud subscriber. This gives the subscriber complete control over role definition, user authorization, and other administrative tasks related to security. Other providers support a multi-tenant model, which pro-vides a predefined environment for the cloud subscriber that is shared with other tenants, typically through tagging data with a subscriber identifier. Tagging gives the appearance of exclusive use of the instance, but relies on the CP to establish and maintain a sound secure database environment.

Data must be secured while at rest, in transit, and in use, and access to the data must be controlled. The client can employ encryption to protect data in transit, though this involves key management responsibilities for the CP. The client can enforce access control techniques but, again, the CP is involved to some extent de-pending on the service model used.

For data at rest, the ideal security measure is for the client to encrypt the data-base and only store encrypted data in the cloud, with the CP having no access to the encryption key. So long as the key remains secure, the CP has no ability to read the data, although corruption and other denial-of-service attacks remain a risk.

A straightforward solution to the security problem in this context is to en-crypt the entire database and not provide the encryption/decryption keys to the service provider. This solution by itself is inflexible. The user has little ability to

Table 16.3 Continued


access individual data items based on searches or indexing on key parameters, but rather would have to download entire tables from the database, decrypt the tables, and work with the results. To provide more flexibility, it must be possible to work with the database in its encrypted form.

An example of such an approach, depicted in Figure 16.10, is reported in [DAMI05] and [DAMI03]. A similar approach is described in [HACI02]. Four enti-ties are involved:

• Data owner: An organization that produces data to be made available for con-trolled release, either within the organization or to external users.

• User: Human entity that presents requests (queries) to the system. The user could be an employee of the organization who is granted access to the data-base via the server, or a user external to the organization who, after authenti-cation, is granted access.

• Client: Frontend that transforms user queries into queries on the encrypted data stored on the server.

• Server: An organization that receives the encrypted data from a data owner and makes them available for distribution to clients. The server could in fact be owned by the data owner but, more typically, is a facility owned and main-tained by an external provider. For our discussion, the server is a cloud server.

Before continuing this discussion, we need to define some database terms. In re-lational database parlance, the basic building block is a relation, which is a flat table. Rows are referred to as tuples, and columns are referred to as attributes. A primary key is defined to be a portion of a row used to uniquely identify a row in a table; the

Queryprocessor

1. Original queryMetadata

4. Plaintextresult

2. Transformedquery

3. Encryptedresult

Client

User

Data owner

Cloudserver

Encrypt/Decrypt

Queryexecutor

Metadata

Metadata

Encrypteddatabase

Database

Figure 16.10 An Encryption Scheme for a Cloud-Based Database

16.7 / Cloud seCurity as a serviCe 517

primary key consists of one or more column names.2 For example, in an employee table, the employee ID is sufficient to uniquely identify a row in a particular table.

Let us first examine the simplest possible arrangement based on this scenario. Suppose that each individual item in the database is encrypted separately, all using the same encryption key. The encrypted database is stored at the server, but the server does not have the encryption key. Thus, the data are secure at the server. Even if someone were able to hack into the server’s system, all he or she would have access to is encrypted data. The client system does have a copy of the encryption key. A user at the client can retrieve a record from the database with the following sequence:

1. The user issues a query for fields from one or more records with a specific value of the primary key.

2. The query processor at the client encrypts the primary key, modifies the query accordingly, and transmits the query to the server.

3. The server processes the query using the encrypted value of the primary key and returns the appropriate record or records.

4. The query processor decrypts the data and returns the results.

This method is certainly straightforward but is quite limited. For example, suppose the Employee table contains a salary attribute and the user wishes to retrieve all records for salaries less than $70K. There is no obvious way to do this, because the attribute value for salary in each record is encrypted. The set of encrypted values does not preserve the ordering of values in the original attribute.

There are a number of ways to extend the functionality of this approach. For example, an unencrypted index value can be associated with a given attribute and the table can be partitioned based on these index values, enabling a user to retrieve a certain portion of the table. The details of such schemes are beyond our scope. See [STAL12] for more detail.

16.7 cLOud security as a service

The term security as a service (SecaaS) has generally meant a package of security services offered by a service provider that offloads much of the security responsibil-ity from an enterprise to the security service provider. Among the services typically provided are authentication, antivirus, antimalware/-spyware, intrusion detection, and security event management. In the context of cloud computing, cloud security as a service, designated SecaaS, is a segment of the SaaS offering of a CP.

The Cloud Security Alliance defines SecaaS as the provision of security ap-plications and services via the cloud either to cloud-based infrastructure and soft-ware or from the cloud to the customers’ on-premise systems [CSA11b]. The Cloud Security Alliance has identified the following SecaaS categories of service:

• Identity and access management

• Data loss prevention

2Note that a primary key has nothing to do with cryptographic keys. A primary key in a database is a means of indexing into the database.


• Web security

• E-mail security

• Security assessments

• Intrusion management

• Security information and event management

• Encryption

• Business continuity and disaster recovery

• Network security

In this section, we examine these categories with a focus on security of the cloud-based infrastructure and services (Figure 16.11).

Identity and access management (IAM) includes people, processes, and sys-tems that are used to manage access to enterprise resources by assuring that the identity of an entity is verified, and then granting the correct level of access based on this assured identity. One aspect of identity management is identity provision-ing, which has to do with providing access to identified users and subsequently

Cloud service clients and adversaries

Identity and access managementNetwork security

Data lossprevention

Web securityIntrusionmanagement

Encryption

E-mail security

Security assessmentsSecurity information and event managementBusiness continuity and disaster recovery

Figure 16.11 Elements of Cloud Security as a Service

16.7 / Cloud seCurity as a serviCe 519

deprovisioning, or deny access, to users when the client enterprise designates such users as no longer having access to enterprise resources in the cloud. Another as-pect of identity management is for the cloud to participate in the federated identity management scheme (see Chapter 15) scheme used by the client enterprise. Among other requirements, the cloud service provider (CSP) must be able to exchange identity attributes with the enterprise’s chosen identity provider.

The access management portion of IAM involves authentication and access control services. For example, the CSP must be able to authenticate users in a trustworthy manner. The access control requirements in SPI environments include establishing trusted user profile and policy information, using it to control access within the cloud service, and doing this in an auditable way.

Data loss prevention (DLP) is the monitoring, protecting, and verifying the security of data at rest, in motion, and in use. Much of DLP can be implemented by the cloud client, such as discussed in Section 16.6. The CSP can also provide DLP services, such as implementing rules about what functions can be performed on data in various contexts.

Web security is real-time protection offered either on premise through soft-ware/appliance installation or via the cloud by proxying or redirecting Web traffic to the CP. This provides an added layer of protection on top of things like antivi-ruses to prevent malware from entering the enterprise via activities such as Web browsing. In addition to protecting against malware, a cloud-based Web security service might include usage policy enforcement, data backup, traffic control, and Web access control.

A CSP may provide a Web-based e-mail service, for which security measures are needed. E-mail security provides control over inbound and outbound e-mail, protecting the organization from phishing, malicious attachments, enforcing corpo-rate polices such as acceptable use and spam prevention. The CSP may also incorpo-rate digital signatures on all e-mail clients and provide optional e-mail encryption.

Security assessments are third-part audits of cloud services. While this service is outside the province of the CSP, the CSP can provide tools and access points to facilitate various assessment activities.

Intrusion management encompasses intrusion detection, prevention, and re-sponse. The core of this service is the implementation of intrusion detection systems (IDSs) and intrusion prevention systems (IPSs) at entry points to the cloud and on servers in the cloud. An IDS is a set of automated tools designed to detect unauthor-ized access to a host system. We discuss this in Chapter 21. An IPS incorporates IDS functionality but also includes mechanisms designed to block traffic from intruders.

Security information and event management (SIEM) aggregates (via push or pull mechanisms) log and event data from virtual and real networks, applications, and systems. This information is then correlated and analyzed to provide real-time reporting and alerting on information/events that may require intervention or other type of response. The CSP typically provides an integrated service that can put to-gether information from a variety of sources both within the cloud and within the client enterprise network.

Encryption is a pervasive service that can be provided for data at rest in the cloud, e-mail traffic, client-specific network management information, and identity information. Encryption services provided by the CSP involve a range of complex


issues, including key management, how to implement virtual private network (VPN) services in the cloud, application encryption, and data content access.

Business continuity and disaster recovery comprise measures and mechanisms to ensure operational resiliency in the event of any service interruptions. This is an area where the CSP, because of economies of scale, can offer obvious benefits to a cloud service client [WOOD10]. The CSP can provide backup at multiple locations, with reliable failover and disaster recovery facilities. This service must include a flexible infrastructure, redundancy of functions and hardware, monitored opera-tions, geographically distributed data centers, and network survivability.

Network security consists of security services that allocate access, distribute, monitor, and protect the underlying resource services. Services include perimeter and server firewalls and denial-of-service protection. Many of the other services listed in this section, including intrusion management, identity and access management, data loss protection, and Web security, also contribute to the network security service.


[NERC11] is a useful overview of the elements of an NAC system. [GEER10] is a concise overview of NAC systems.

[HOEP09] is an excellent introduction to EAP. [CHEN05b] provides a good overview of EAP and 802.1X. [CHEN05a] provides more detailed coverage of 802.1X.

[JANS11] is a worthwhile, systematic treatment of cloud security issues. Other useful treat-ments, providing differing perspectives, are [HASS10], [BALA09], [ANTH10], and [CSA11a].

ANTH10 Anthes, G. “Security in the Cloud.” Communications of the ACM, November 2010.

BALA09 Balachandra, R.; Ramakrishna, P.; and Rakshit, A. “Cloud Security Issues.” Proceedings, 2009 IEEE International Conference on Services Computing, 2009.

CHEN05a Chen, J.; Jiang, M.; and Liu, Y. “Wireless LAN Security and IEEE 802.i.” IEEE Wireless Communications, February 2005.

CHEN05b Chen, J., and Wang, Y. “Extensible Authentication Protocol (EAP) and IEEE 802.1x: Tutorial and Empirical Experience.” IEEE Radio Communications, December 2005.

CSA11a Cloud Security Alliance. Security Guidance for Critical Areas of Focus in Cloud Computing V2.1. CSA Report, 2011.

GEER10 Geer, D. “Whatever Happened to Network-Access-Control Technology?” Computer, September 2010.

HASS10 Hassan, T.; Joshi, J.; and Ahn, G. “Security and Privacy Challenges in Cloud Computing Environments.” IEEE Security & Privacy, November/December 2010.

HOEP09 Hoeper, K., and Chen, L. Recommendation for EAP Methods Used in Wireless Network Access Authentication. NIST Special Publication 800-120, September 2009.

JANS11 Jansen, W., and Grance, T. Guidelines on Security and Privacy in Public Cloud Computing. NIST Special Publication 800-144, January 2011.

NERC11 North American Electric Reliability Corp. Guidance for Secure Interactive Remote Access. July 2011. www.nerc.com

www.nerc.com

16.9 / key terms, review QuestioNs, aNd problems 521

16.9 key terms, review QuestiOns, and PrObLems

Key Terms

access requestor (AR)authentication servercloudcloud auditorcloud brokercloud carriercloud computingcloud consumercloud providercommunity cloudDynamic Host Configuration

Protocol (DHCP)EAP authenticator

EAP-GPSKEAP-IKEv2EAP over LAN (EAPOL)EAP methodEAP pass-through modeEAP peerEAP-TLSEAP-TTLSExtensible Authentication

Protocol (EAP)firewallIEEE 802.1Xmedia gateway

network access control (NAC)

network access server (NAS)platform as a service (PaaS)policy serverprivate cloudpublic cloudremote access server (RAS)security as a service (SecaaS)software as a service (SaaS)supplicantvirtual local area network

(VLAN)

Review Questions 16.1 Provide a brief definition of network access control. 16.2 What is an EAP? 16.3 List and briefly define four EAP authentication methods. 16.4 What is EAPOL? 16.5 What is the function of IEEE 802.1X? 16.6 Define cloud computing. 16.7 List and briefly define three cloud service models. 16.8 What is the cloud computing reference architecture? 16.9 Describe some of the main cloud-specific security threats.

Problems 16.1 Investigate the network access control scheme used at your school or place of em-

ployment. Draw a diagram and describe the principal components. 16.2 Figure 16.3 suggests that EAP can be described in the context of a four-layer model.

Indicate the functions and formats of each of the four layers. You may need to refer to RFC 3748.

16.3 Find and view several YouTube videos that discuss cloud security. Identify the URLs of three videos that you think do a good job communicating the essential issues and approaches for cloud security. If you could only recommend one to fellow students, which would you pick? Why? Summarize your recommendations and justification in a brief paper (250–500 words) or a three- to five-slide PowerPoint presentation.

TransporT-LeveL securiTy17.1 Web Security Considerations

Web Security ThreatsWeb Traffic Security Approaches

17.2 Secure Sockets Layer

SSL ArchitectureSSL Record ProtocolChange Cipher Spec ProtocolAlert ProtocolHandshake ProtocolCryptographic Computations

17.3 Transport Layer Security

Version Number Message Authentication CodePseudorandom FunctionAlert CodesCipher SuitesClient Certificate TypesCertificate_Verify and Finished MessagesCryptographic ComputationsPadding

17.4 HTTPS

Connection InitiationConnection Closure

17.5 Secure Shell (SSH)

Transport Layer ProtocolUser Authentication ProtocolConnection Protocol



Chapter

522522

17.1 / Web Security conSiderationS 523

Learning Objectives


u Summarize Web security threats and Web traffic security approaches.

u Present an overview of Secure Sockets Layer (SSL).

u Understand the differences between Secure Sockets Layer and Transport Layer Security.

u Compare the pseudorandom function used in Transport Layer Security with those discussed earlier in the book.

u Present an overview of HTTPS (HTTP over SSL).

u Present an overview of Secure Shell (SSH).

Virtually all businesses, most government agencies, and many individuals now have Web sites. The number of individuals and companies with Internet access is expanding rapidly and all of these have graphical Web browsers. As a result, businesses are enthu-siastic about setting up facilities on the Web for electronic commerce. But the reality is that the Internet and the Web are extremely vulnerable to compromises of various sorts. As businesses wake up to this reality, the demand for secure Web services grows.

The topic of Web security is a broad one and can easily fill a book. In this chapter, we begin with a discussion of the general requirements for Web security and then focus on three standardized schemes that are becoming increasingly important as part of Web commerce and that focus on security at the transport layer: SSL/TLS, HTTPS, and SSH.

17.1 Web security cOnsideratiOns

The World Wide Web is fundamentally a client/server application running over the Internet and TCP/IP intranets. As such, the security tools and approaches discussed so far in this book are relevant to the issue of Web security. However, the following characteristics of Web usage suggest the need for tailored security tools:

• Although Web browsers are very easy to use, Web servers are relatively easy to configure and manage, and Web content is increasingly easy to develop, the underlying software is extraordinarily complex. This complex software may hide many potential security flaws. The short history of the Web is filled with examples of new and upgraded systems, properly installed, that are vulnerable to a variety of security attacks.

We cannot enter into alliance with neighboring princes until we are acquainted with their designs.


524 chapter 17 / tranSport-LeveL Security

• A Web server can be exploited as a launching pad into the corporation’s or agency’s entire computer complex. Once the Web server is subverted, an at-tacker may be able to gain access to data and systems not part of the Web itself but connected to the server at the local site.

• Casual and untrained (in security matters) users are common clients for Web-based services. Such users are not necessarily aware of the security risks that exist and do not have the tools or knowledge to take effective countermeasures.

Web Security Threats

Table 17.1 provides a summary of the types of security threats faced when using the Web. One way to group these threats is in terms of passive and active attacks. Passive attacks include eavesdropping on network traffic between browser and server and gaining access to information on a Web site that is supposed to be re-stricted. Active attacks include impersonating another user, altering messages in transit between client and server, and altering information on a Web site.

Another way to classify Web security threats is in terms of the location of the threat: Web server, Web browser, and network traffic between browser and server. Issues of server and browser security fall into the category of computer system security; Part Six of this book addresses the issue of system security in general but is also applicable to Web system security. Issues of traffic security fall into the category of network security and are addressed in this chapter.

Table 17.1 A Comparison of Threats on the Web

Threats Consequences Countermeasures

Integrity •Modificationofuserdata

•Trojanhorsebrowser

•Modificationofmemory

•Modificationofmessage traffic in transit

•Lossofinformation

•Compromiseofmachine

•Vulnerabilitytoallotherthreats

Cryptographic checksums

Confidentiality •Eavesdroppingonthenet

•Theftofinfofromserver

•Theftofdatafromclient

•Infoaboutnetwork configuration

•Infoaboutwhichclienttalksto server

•Lossofinformation

•Lossofprivacy

Encryption,Webproxies

Denial of Service

•Killingofuserthreads

•Floodingmachinewithbogusrequests

•Fillingupdiskormemory

•IsolatingmachinebyDNSattacks

•Disruptive

•Annoying

•Preventuserfromgettingwork done

Difficulttoprevent

Authentication •Impersonationoflegitimateusers

•Dataforgery

•Misrepresentationofuser

•Beliefthatfalseinformationis valid

Cryptographic techniques

17.2 / Secure SocketS Layer 525

Web Traffic Security Approaches

A number of approaches to providing Web security are possible. The various ap-proaches that have been considered are similar in the services they provide and, to some extent, in the mechanisms that they use, but they differ with respect to their scope of applicability and their relative location within the TCP/IP protocol stack.

Figure 17.1 illustrates this difference. One way to provide Web security is to use IP security (IPsec) (Figure 17.1a). The advantage of using IPsec is that it is transparent to end users and applications and provides a general-purpose solution. Furthermore, IPsec includes a filtering capability so that only selected traffic need incur the overhead of IPsec processing.

Another relatively general-purpose solution is to implement security justabove TCP (Figure 17.1b). The foremost example of this approach is the Secure Sockets Layer (SSL) and the follow-on Internet standard known as Transport Layer Security (TLS). At this level, there are two implementation choices. For full generality, SSL (or TLS) could be provided as part of the underlying protocol suite and therefore be transparent to applications. Alternatively, SSL can be embedded inspecificpackages.Forexample,NetscapeandMicrosoftExplorerbrowserscomeequipped with SSL, and most Web servers have implemented the protocol.

Application-specific security services are embedded within the particular application. Figure 17.1c shows examples of this architecture. The advantage of this approach is that the service can be tailored to the specific needs of a given application.

17.2 secure sOckets Layer

One of the most widely used security services is the Secure Sockets Layer (SSL) and the follow-on Internet standard known as Transport Layer Security (TLS), the lat-ter defined in RFC 5246. SSL is a general-purpose service implemented as a set of protocols that rely on TCP. At this level, there are two implementation choices. For full generality, SSL (or TLS) could be provided as part of the underlying protocol suite and therefore be transparent to applications. Alternatively, SSL can be em-bedded in specific packages. For example, most browsers come equipped with SSL, and most Web servers have implemented the protocol.

This section is devoted to a discussion of SSLv3, and next section describes the principal differences between SSLv3 and TLS.

SMTPHTTP

TCP

IP/IPSec

(a) Network level

FTP

SMTPHTTP

TCP

SSL or TLS

IP

(b) Transport level

FTP

IP

S/MIME

HTTPKerberos

UDP

SMTP

(c) Application level

TCP

Figure 17.1 Relative Location of Security Facilities in the TCP/IP Protocol Stack


SSL Architecture

SSL is designed to make use of TCP to provide a reliable end-to-end secure ser-vice. SSL is not a single protocol but rather two layers of protocols, as illustrated in Figure 17.2.

The SSL Record Protocol provides basic security services to various higher-layer protocols. In particular, the Hypertext Transfer Protocol (HTTP), which provides the transfer service for Web client/server interaction, can operate on top of SSL. Three higher-layer protocols are defined as part of SSL: the Handshake Protocol, The Change Cipher Spec Protocol, and the Alert Protocol. These SSL-specific protocols are used in the management of SSL exchanges and are examined later in this section.

Two important SSL concepts are the SSL session and the SSL connection, which are defined in the specification as follows.

• Connection: A connection is a transport (in the OSI layering model defini-tion) that provides a suitable type of service. For SSL, such connections are peer-to-peerrelationships.Theconnectionsaretransient.Everyconnectionisassociated with one session.

• Session: An SSL session is an association between a client and a server. Sessions are created by the Handshake Protocol. Sessions define a set of cryp-tographic security parameters which can be shared among multiple connec-tions. Sessions are used to avoid the expensive negotiation of new security parameters for each connection.

Between any pair of parties (applications such as HTTP on client and server), there may be multiple secure connections. In theory, there may also be multiple simultaneous sessions between parties, but this feature is not used in practice.

There are a number of states associated with each session. Once a session is es-tablished, there is a current operating state for both read and write (i.e., receive and send). In addition, during the Handshake Protocol, pending read and write states are created. Upon successful conclusion of the Handshake Protocol, the pending states become the current states.

IP

TCP

SSL Record Protocol

SSLHandshake

Protocol

SSL ChangeCipher Spec

Protocol

SSL AlertProtocol

HTTP

Figure 17.2 SSL Protocol Stack


A session state is defined by the following parameters.

• Session identifier: An arbitrary byte sequence chosen by the server to identify an active or resumable session state.

• Peer certificate: An X509.v3 certificate of the peer. This element of the state may be null.

• Compression method: The algorithm used to compress data prior to encryption.

• Cipher spec: Specifiesthebulkdataencryptionalgorithm(suchasnull,AES,etc.)andahashalgorithm(suchasMD5orSHA-1)usedforMACcalcula-tion. It also defines cryptographic attributes such as the hash_size.

• Master secret: 48-byte secret shared between the client and the server.

• Is resumable: A flag indicating whether the session can be used to initiate new connections.

A connection state is defined by the following parameters.

• Server and client random: Byte sequences that are chosen by the server and client for each connection.

• Server write MAC secret: The secret key used in MAC operations on data sent by the server.

• Client write MAC secret: The secret key used in MAC operations on data sent by the client.

• Server write key: The secret encryption key for data encrypted by the server and decrypted by the client.

• Client write key: The symmetric encryption key for data encrypted by the cli-ent and decrypted by the server.

• Initialization vectors: When a block cipher in CBC mode is used, an initial-ization vector (IV) is maintained for each key. This field is first initialized by the SSL Handshake Protocol. Thereafter, the final ciphertext block from each record is preserved for use as the IV with the following record.

• Sequence numbers: Eachpartymaintains separate sequencenumbers fortransmitted and received messages for each connection. When a party sends or receives a change cipher spec message, the appropriate sequence number is set to zero. Sequence numbers may not exceed 264 - 1.

SSL Record Protocol

The SSL Record Protocol provides two services for SSL connections:

• Confidentiality: The Handshake Protocol defines a shared secret key that is used for conventional encryption of SSL payloads.

• Message Integrity: The Handshake Protocol also defines a shared secret key that is used to form a message authentication code (MAC).


Figure 17.3 indicates the overall operation of the SSL Record Protocol. The Record Protocol takes an application message to be transmitted, fragments the data into manageable blocks, optionally compresses the data, applies a MAC, encrypts, adds a header, and transmits the resulting unit in a TCP segment. Received data are decrypted, verified, decompressed, and reassembled before being delivered to higher-level users.

The first step is fragmentation.Eachupper-layermessageisfragmentedintoblocks of 214 bytes (16384 bytes) or less. Next, compression is optionally applied. Compression must be lossless and may not increase the content length by more than 1024 bytes.1 In SSLv3 (as well as the current version of TLS), no compression algo-rithm is specified, so the default compression algorithm is null.

The next step in processing is to compute a message authentication code over the compressed data. For this purpose, a shared secret key is used. The calculation is defined as

hash(MAC_write_secret } pad_2 } hash(MAC_write_secret } pad_1 } seq_num } SSLCompressed.type } SSLCompressed.length } SSLCompressed.fragment))

where

} = concatenation

MAC_write_secret = shared secret key

hash = cryptographichashalgorithm;eitherMD5or SHA-1

Application data

Fragment

Compress

Add MAC

Encrypt

Append SSLrecord header

Figure 17.3 SSL Record Protocol Operation

1Of course, one hopes that compression shrinks rather than expands the data. However, for very short blocks, it is possible, because of formatting conventions, that the compression algorithm will actually pro-vide output that is longer than the input.


pad_1 = the byte 0x36 (0011 0110) repeated 48times(384bits)forMD5and40times(320 bits) for SHA-1

pad_2 = the byte 0x5C (0101 1100) repeated 48 timesforMD5and40timesforSHA-1

seq_num = the sequence number for this messageSSLCompressed.type = the higher-level protocol used to process

this fragmentSSLCompressed.length = the length of the compressed fragmentSSLCompressed.fragment = the compressed fragment (if compression

is not used, this is the plaintext fragment)

Note that this is very similar to the HMAC algorithm defined in Chapter 12. The difference is that the two pads are concatenated in SSLv3 and are XORed in HMAC. The SSLv3 MAC algorithm is based on the original Internet draft for HMAC, which used concatenation. The final version of HMAC (defined in RFC 2104) uses the XOR.

Next, the compressed message plus the MAC are encrypted using symmet-ricencryption.Encryptionmaynotincreasethecontentlengthbymorethan1024bytes, so that the total length may not exceed 214 + 2048. The following encryption algorithms are permitted:

Block Cipher Stream Cipher

Algorithm Key Size Algorithm Key Size

AES 128, 256 RC4-40 40

IDEA 128 RC4-128 128

RC2-40 40

DES-40 40

DES 56

3DES 168

Fortezza 80

Fortezza can be used in a smart card encryption scheme.For stream encryption, the compressed message plus the MAC are encrypted.

Note that the MAC is computed before encryption takes place and that the MAC is then encrypted along with the plaintext or compressed plaintext.

For block encryption, padding may be added after the MAC prior to encryp-tion. The padding is in the form of a number of padding bytes followed by a one-byte indication of the length of the padding. The total amount of padding is the smallest amount such that the total size of the data to be encrypted (plaintext plus MAC plus padding) is a multiple of the cipher’s block length. An example is a plain-text (or compressed text if compression is used) of 58 bytes, with a MAC of 20 bytes (usingSHA-1),thatisencryptedusingablocklengthof8bytes(e.g.,DES).Withthe padding-length byte, this yields a total of 79 bytes. To make the total an integer multiple of 8, one byte of padding is added.


The final step of SSL Record Protocol processing is to prepare a header con-sisting of the following fields:

• Content Type (8 bits): The higher-layer protocol used to process the enclosed fragment.

• Major Version (8 bits): IndicatesmajorversionofSSLinuse.ForSSLv3,thevalue is 3.

• Minor Version (8 bits): Indicates minor version in use. For SSLv3, the value is 0.

• Compressed Length (16 bits): The length in bytes of the plaintext fragment (or compressed fragment if compression is used). The maximum value is 214 + 2048.

The content types that have been defined are change_cipher_spec, alert, handshake, and application_data. The first three are the SSL-specific protocols, discussed next. Note that no distinction is made among the various appli-cations (e.g., HTTP) that might use SSL; the content of the data created by such applications is opaque to SSL.

Figure 17.4 illustrates the SSL record format.

Change Cipher Spec Protocol

The Change Cipher Spec Protocol is one of the three SSL-specific protocols that use the SSL Record Protocol, and it is the simplest. This protocol consists of a single message (Figure 17.5a), which consists of a single byte with the value 1. The sole purpose of this message is to cause the pending state to be copied into the current state, which updates the cipher suite to be used on this connection.

Alert Protocol

The Alert Protocol is used to convey SSL-related alerts to the peer entity. As with other applications that use SSL, alert messages are compressed and encrypted, as specified by the current state.

Contenttype

Majorversion

Minorversion

Compressedlength

Plaintext(optionally

compressed)

MAC (0, 16, or 20 bytes)

Enc

rypt

ed

Figure 17.4 SSL Record Format


Eachmessageinthisprotocolconsistsoftwobytes(Figure17.5b).Thefirstbyte takes the value warning (1) or fatal (2) to convey the severity of the message. If the level is fatal, SSL immediately terminates the connection. Other connec-tions on the same session may continue, but no new connections on this session may be established. The second byte contains a code that indicates the specific alert. First, we list those alerts that are always fatal (definitions from the SSL specification):

• unexpected_message: An inappropriate message was received.

• bad_record_mac: An incorrect MAC was received.

• decompression_failure: The decompression function received im-proper input (e.g., unable to decompress or decompress to greater than maxi-mum allowable length).

• handshake_failure: Sender was unable to negotiate an acceptable set of security parameters given the options available.

• illegal_parameter: A field in a handshake message was out of range or inconsistent with other fields.

The remaining alerts are the following.

• close_notify: Notifies the recipient that the sender will not send any moremessagesonthisconnection.Eachpartyisrequiredtosendaclose_ notify alert before closing the write side of a connection.

• no_certificate: May be sent in response to a certificate request if no ap-propriate certificate is available.

• bad_certificate: A received certificate was corrupt (e.g., contained a signature that did not verify).

• unsupported_certificate: The type of the received certificate is not supported.

• certificate_revoked: A certificate has been revoked by its signer.

• certificate_expired: A certificate has expired.

• certificate_unknown: Some other unspecified issue arose in processing the certificate, rendering it unacceptable.

1

(a) Change Cipher Spec Protocol

1 byte

Type

(c) Handshake Protocol

1 byte

Length

3 bytes

Content

� 0 bytes

(d) Other Upper-Layer Protocol (e.g., HTTP)

Opaque content

� 1 byte

Level

(b) Alert Protocol

1 byte 1 byte

Alert

Figure 17.5 SSL Record Protocol Payload


Handshake Protocol

The most complex part of SSL is the Handshake Protocol. This protocol allows the server and client to authenticate each other and to negotiate an encryption and MAC algorithm and cryptographic keys to be used to protect data sent in an SSL record. The Handshake Protocol is used before any application data is transmitted.

The Handshake Protocol consists of a series of messages exchanged by client andserver.AllofthesehavetheformatshowninFigure17.5c.Eachmessagehasthree fields:

• Type (1 byte): Indicates one of ten messages. Table 17.2 lists the defined message types.

• Length (3 bytes): The length of the message in bytes.

• Content (# 0 bytes): The parameters associated with this message; these are listed in Table 17.2.

Figure 17.6 shows the initial exchange needed to establish a logical connection between client and server. The exchange can be viewed as having four phases.

Phase 1. establish security caPabilities This phase is used to initiate a logical connection and to establish the security capabilities that will be associated with it. The exchange is initiated by the client, which sends a client_hello message with the following parameters:

• Version: The highest SSL version understood by the client.

• Random: A client-generated random structure consisting of a 32-bit time-stamp and 28 bytes generated by a secure random number generator. These values serve as nonces and are used during key exchange to prevent replay attacks.

• Session ID: A variable-length session identifier. A nonzero value indicates that the client wishes to update the parameters of an existing connection or to

Table 17.2 SSL Handshake Protocol Message Types

Message Type Parameters

hello_request null

client_hello version, random, session id, cipher suite, compression method

server_hello version, random, session id, cipher suite, compression method

certificate chain of X.509v3 certificates

server_key_exchange parameters, signature

certificate_request type, authorities

server_done null

certificate_verify signature

client_key_exchange parameters, signature

finished hash value


create a new connection on this session. A zero value indicates that the client wishes to establish a new connection on a new session.

• CipherSuite: This is a list that contains the combinations of cryptographic al-gorithmssupportedbytheclient,indecreasingorderofpreference.Eachele-ment of the list (each cipher suite) defines both a key exchange algorithm and a CipherSpec; these are discussed subsequently.

• Compression Method: This is a list of the compression methods the client supports.

Client Server

Phase 1Establish security capabilities, includingprotocol version, session ID, cipher suite,compression method, and initial randomnumbers.

Phase 2Server may send certificate, key exchange,and request certificate. Server signals endof hello message phase.

Phase 3Client sends certificate if requested. Clientsends key exchange. Client may sendcertificate verification.

Phase 4Change cipher suite and finishhandshake protocol.

Note: Shaded transfers areoptional or situation-dependentmessages that are not always sent.

finished

change_cipher_spec

finished

change_cipher_spec

certificate_verify

client_key_exchange

certificate

server_hello_done

certificate_request

server_key_exchange

certificate

server_hello

client_hello

Tim

e

Figure 17.6 Handshake Protocol Action


After sending the client_hello message, the client waits for the server_hello message, which contains the same parameters as the client_hello message. For the server_hello message, the following conventions apply. The Version field contains the lower of the versions suggested by the client and the high-est supported by the server. The Random field is generated by the server and is independentoftheclient’sRandomfield.IftheSessionIDfieldoftheclientwasnonzero,thesamevalueisusedbytheserver;otherwisetheserver’sSessionIDfieldcontains the value for a new session. The CipherSuite field contains the single cipher suite selected by the server from those proposed by the client. The Compression field contains the compression method selected by the server from those proposed by the client.

The first element of the CipherSuite parameter is the key exchange method (i.e., the means by which the cryptographic keys for conventional encryption and MAC are exchanged). The following key exchange methods are supported.

• RSA: The secret key is encrypted with the receiver’s RSA public key. A pub-lic-key certificate for the receiver’s key must be made available.

• Fixed Diffie-Hellman: This isaDiffie-Hellmankeyexchangeinwhichtheserver’scertificatecontainstheDiffie-Hellmanpublicparameterssignedbythe certificate authority (CA). That is, the public-key certificate contains the Diffie-Hellmanpublic-keyparameters.TheclientprovidesitsDiffie-Hellmanpublic-key parameters either in a certificate, if client authentication is re-quired, or in a key exchange message. This method results in a fixed secret key betweentwopeersbasedontheDiffie-Hellmancalculationusingthefixedpublic keys.

• Ephemeral Diffie-Hellman: This technique is used to create ephemeral (tem-porary,one-time)secretkeys.Inthiscase,theDiffie-Hellmanpublickeysareexchanged,signedusingthesender’sprivateRSAorDSSkey.Thereceivercan use the corresponding public key to verify the signature. Certificates are used to authenticate the public keys. This would appear to be the most secure ofthethreeDiffie-Hellmanoptions,becauseitresultsinatemporary,authen-ticated key.

• Anonymous Diffie-Hellman: The baseDiffie-Hellman algorithm is usedwithnoauthentication.Thatis,eachsidesendsitspublicDiffie-Hellmanpa-rameters to the other with no authentication. This approach is vulnerable to man-in-the-middleattacks,inwhichtheattackerconductsanonymousDiffie-Hellman with both parties.

• Fortezza: The technique defined for the Fortezza scheme.

Following the definition of a key exchange method is the CipherSpec, which includes the following fields.

• CipherAlgorithm: Anyofthealgorithmsmentionedearlier:RC4,RC2,DES,3DES,DES40,IDEA,orFortezza

• MACAlgorithm: MD5orSHA-1

• CipherType: Stream or Block


• IsExportable: True or False

• HashSize: 0,16(forMD5),or20(forSHA-1)bytes

• Key Material: A sequence of bytes that contain data used in generating the write keys

• IV Size: The size of the Initialization Value for Cipher Block Chaining (CBC) encryption

Phase 2. server authentication and Key exchange The server begins this phase by sending its certificate if it needs to be authenticated; the message con-tains one or a chain of X.509 certificates. The certificate message is required for anyagreed-onkeyexchangemethodexceptanonymousDiffie-Hellman.NotethatiffixedDiffie-Hellmanisused,thiscertificatemessagefunctionsastheserver’skey exchangemessage because it contains the server’s publicDiffie-Hellmanparameters.

Next, a server_key_exchange message may be sent if it is required. It is notrequiredintwoinstances:(1)TheserverhassentacertificatewithfixedDiffie-Hellman parameters or (2) a RSA key exchange is to be used. The server_key_exchange message is needed for the following:

• Anonymous Diffie-Hellman: The message content consists of the two global Diffie-Hellmanvalues(aprimenumberandaprimitiverootofthatnumber)plustheserver’spublicDiffie-Hellmankey(see Figure 10.1).

• Ephemeral Diffie-Hellman: ThemessagecontentincludesthethreeDiffie-HellmanparametersprovidedforanonymousDiffie-Hellmanplusasignatureof those parameters.

• RSA key exchange (in which the server is using RSA but has a signature-only RSA key): Accordingly, the client cannot simply send a secret key encrypted with the server’s public key. Instead, the server must create a temporary RSA public/private key pair and use the server_key_exchange message to send the public key. The message content includes the two parameters of the tem-porary RSA public key (exponent and modulus; see Figure 9.5) plus a signa-ture of those parameters.

• Fortezza

Some further details about the signatures are warranted. As usual, a signature is created by taking the hash of a message and encrypting it with the sender’s private key. In this case, the hash is defined as

hash(ClientHello.random } ServerHello.random } ServerParams)

SothehashcoversnotonlytheDiffie-HellmanorRSAparametersbutalsothetwo nonces from the initial hello messages. This ensures against replay attacks and misrepresentation.InthecaseofaDSSsignature,thehashisperformedusingtheSHA-1algorithm.InthecaseofanRSAsignature,bothanMD5andanSHA-1hash are calculated, and the concatenation of the two hashes (36 bytes) is encrypted with the server’s private key.


Next,anonanonymousserver(servernotusinganonymousDiffie-Hellman)can request a certificate from the client. The certificate_request message includes two parameters: certificate_type and certificate_authorities. The certificate type indicates the public-key algorithm and its use:

• RSA, signature only

• DSS,signatureonly

• RSAforfixedDiffie-Hellman;inthiscasethesignatureisusedonlyforau-thentication, by sending a certificate signed with RSA

• DSSforfixedDiffie-Hellman;again,usedonlyforauthentication

• RSAforephemeralDiffie-Hellman

• DSSforephemeralDiffie-Hellman

• Fortezza

The second parameter in the certificate_request message is a list of the distinguished names of acceptable certificate authorities.

The final message in phase 2, and one that is always required, is the server_done message, which is sent by the server to indicate the end of the server hello and associated messages. After sending this message, the server will wait for a client response. This message has no parameters.

Phase 3. client authentication and Key exchange Upon receipt of the server_done message, the client should verify that the server provided a valid certificate (if required) and check that the server_hello parameters are acceptable. If all is satisfactory, the client sends one or more messages back to the server.

If the server has requested a certificate, the client begins this phase by send-ing a certificate message. If no suitable certificate is available, the client sends a no_certificate alert instead.

Next is the client_key_exchange message, which must be sent in this phase. The content of the message depends on the type of key exchange, as follows.

• RSA: The client generates a 48-byte pre-master secret and encrypts with the public key from the server’s certificate or temporary RSA key from a server_key_exchange message. Its use to compute a master secret is ex-plained later.

• Ephemeral or Anonymous Diffie-Hellman: Theclient’spublicDiffie-Hellmanparameters are sent.

• Fixed Diffie-Hellman: Theclient’spublicDiffie-Hellmanparametersweresent in a certificate message, so the content of this message is null.

• Fortezza: The client’s Fortezza parameters are sent.

Finally, in this phase, the client may send a certificate_verify message to provide explicit verification of a client certificate. This message is only sent fol-lowing any client certificate that has signing capability (i.e., all certificates except


thosecontainingfixedDiffie-Hellmanparameters).Thismessagesignsahashcodebased on the preceding messages, defined as

CertificateVerify.signature.md5_hash=

MD5( master_secret } pad_2 } MD5(handshake_messages } master_secret } pad_1));

CertificateVerify.signature.sha_hash=

SHA( master_secret } pad_2 } SHA(handshake_messages } master_secret } pad_1));

where pad_1 and pad_2 are the values defined earlier for the MAC, handshake_messages refers to all Handshake Protocol messages sent or received starting at client_hello but not including this message, and master_secret is the calcu-lated secret whose construction is explained later in this section. If the user’s private keyisDSS,thenitisusedtoencrypttheSHA-1hash.Iftheuser’sprivatekeyisRSA,it isusedtoencrypttheconcatenationoftheMD5andSHA-1hashes.Ineither case, the purpose is to verify the client’s ownership of the private key for theclientcertificate.Evenifsomeoneismisusingtheclient’scertificate,heorshewould be unable to send this message.

Phase 4. Finish This phase completes the setting up of a secure connection. The client sends a change_cipher_spec message and copies the pending CipherSpec into the current CipherSpec. Note that this message is not considered part of the Handshake Protocol but is sent using the Change Cipher Spec Protocol. The client then imme-diately sends the finished message under the new algorithms, keys, and secrets. The finished message verifies that the key exchange and authentication processes were suc-cessful. The content of the finished message is the concatenation of two hash values:

MD5( master_secret } pad2 } MD5(handshake_messages } Sender } master_secret } pad1))

SHA( master_secret } pad2 } SHA(handshake_messages } Sender } master_secret } pad1))

where Sender is a code that identifies that the sender is the client and handshake_messages is all of the data from all handshake messages up to but not including this message.

In response to these two messages, the server sends its own change_ cipher_spec message, transfers the pending to the current CipherSpec, and sends its finished message. At this point, the handshake is complete and the client and server may begin to exchange application-layer data.

Cryptographic Computations

Two further items are of interest: (1) the creation of a shared master secret by means of the key exchange and (2) the generation of cryptographic parameters from the master secret.


Master secret creation The shared master secret is a one-time 48-byte value (384 bits) generated for this session by means of secure key exchange. The creation is in two stages. First, a pre_master_secret is exchanged. Second, the master_secret is calculated by both parties. For pre_master_secret exchange, there are two possibilities.

• RSA: A 48-byte pre_master_secret is generated by the client, encrypted with the server’s public RSA key, and sent to the server. The server decrypts the ciphertext using its private key to recover the pre_master_secret.

• Diffie-Hellman: BothclientandservergenerateaDiffie-Hellmanpublickey.Aftertheseareexchanged,eachsideperformstheDiffie-Hellmancalculationto create the shared pre_master_secret.

Both sides now compute the master_secret as

master_secret = MD5( pre_master_secret } SHA('A' } pre_master_secret } ClientHello.random } ServerHello.random)) }

MD5( pre_master_secret } SHA('BB' } pre_master_secret } ClientHello.random } ServerHello.random)) }

MD5( pre_master_secret } SHA('CCC' } pre_master_secret } ClientHello.random } ServerHello.random))

where ClientHello.random and ServerHello.random are the two nonce values exchanged in the initial hello messages.

generation oF cryPtograPhic ParaMeters CipherSpecs require a client write MAC secret, a server write MAC secret, a client write key, a server write key, a client write IV, and a server write IV, which are generated from the master secret in that order. These parameters are generated from the master secret by hashing the master secret into a sequence of secure bytes of sufficient length for all needed parameters.

The generation of the key material from the master secret uses the same for-mat for generation of the master secret from the pre-master secret as

key_block = MD5( master_secret } SHA('A' } master_secret } ServerHello.random } ClientHello.random)) }

MD5( master_secret } SHA('BB' } master_secret } ServerHello.random } ClientHello.random)) }

MD5( master_secret } SHA('CCC' } master_secret } ServerHello.random } ClientHello.random)) } . . .

until enough output has been generated. The result of this algorithmic structure is a pseudorandom function. We can view the master_secret as the pseudorandom seed value to the function. The client and server random numbers can be viewed as salt values to complicate cryptanalysis (see Chapter 22 for a discussion of the use of salt values).

17.3 / tranSport Layer Security 539

17.3 transpOrt Layer security

TLSisanIETFstandardizationinitiativewhosegoalistoproduceanInternetstan-dard version of SSL. TLS is defined as a Proposed Internet Standard in RFC 5246. RFC 5246 is very similar to SSLv3. In this section, we highlight the differences.

Version Number

The TLS Record Format is the same as that of the SSL Record Format (Figure 17.4), and the fields in the header have the same meanings. The one difference is in version values.ForthecurrentversionofTLS,themajorversionis3andtheminorversionis3.

Message Authentication Code

There are two differences between the SSLv3 and TLS MAC schemes: the actual algorithm and the scope of the MAC calculation. TLS makes use of the HMAC algorithm defined in RFC 2104. Recall from Chapter 12 that HMAC is defined as

HMACK(M) = H[(K+ ⊕ opad) } H[(K+ ⊕ ipad) } M]]

where

H = embeddedhashfunction(forTLS,eitherMD5orSHA-1)

M = message input to HMAC

K+ = secret key padded with zeros on the left so that the result is equal to theblocklengthofthehashcode(forMD5andSHA-1,blocklength= 512 bits)

ipad = 00110110 (36 in hexadecimal) repeated 64 times (512 bits)opad = 01011100 (5C in hexadecimal) repeated 64 times (512 bits)

SSLv3 uses the same algorithm, except that the padding bytes are concate-nated with the secret key rather than being XORed with the secret key padded to the block length. The level of security should be about the same in both cases.

For TLS, the MAC calculation encompasses the fields indicated in the following expression:

MAC(MAC_write_secret,seq_num } TLSCompressed.type } TLSCompressed.version } TLSCompressed.length } TLSCompressed.fragment)

The MAC calculation covers all of the fields covered by the SSLv3 calcula-tion, plus the field TLSCompressed.version, which is the version of the protocol being employed.

Pseudorandom Function

TLS makes use of a pseudorandom function referred to as PRF to expand secrets intoblocksofdataforpurposesofkeygenerationorvalidation.Theobjectiveistomake use of a relatively small shared secret value but to generate longer blocks of


data in a way that is secure from the kinds of attacks made on hash functions and MACs. The PRF is based on the data expansion function (Figure 17.7) given as

P_hash(secret, seed)= HMAC_hash(secret,A(1) } seed) } HMAC_hash(secret,A(2) } seed) } HMAC_hash(secret,A(3) } seed) } . . .

where A() is defined as

A(0) = seedA(i) = HMAC_hash(secret, A(i - 1))

ThedataexpansionfunctionmakesuseoftheHMACalgorithmwitheitherMD5or SHA-1 as the underlying hash function. As can be seen, P_hash can be iterated as many times as necessary to produce the required quantity of data. For example, if P_SHA-1 was used to generate 64 bytes of data, it would have to be iterated four times, producing 80 bytes of data of which the last 16 would be discarded. In this

Secret

Seed

Seed

A(1)HMAC

Secret

Secret

Length � hash size

Secret

Seed

A(2)HMAC

HMAC Secret

Seed

A(3)HMAC

HMAC

Secret HMAC

Figure 17.7 TLS Function P_hash(secret, seed)

17.3 / tranSport Layer Security 541

case, P_MD5 would also have to be iterated four times, producing exactly 64 bytes of data. Note that each iteration involves two executions of HMAC—each of which in turn involves two executions of the underlying hash algorithm.

To make PRF as secure as possible, it uses two hash algorithms in a way that should guarantee its security if either algorithm remains secure. PRF is defined as

PRF(secret, label, seed) = P_hash(S1, label } seed)

PRF takes as input a secret value, an identifying label, and a seed value and produces an output of arbitrary length.

Alert Codes

TLS supports all of the alert codes defined in SSLv3 with the exception of no_certificate. A number of additional codes are defined in TLS; of these, the following are always fatal.

• record_overflow: A TLS record was received with a payload (ciphertext) whose length exceeds 214 + 2048 bytes, or the ciphertext decrypted to a length of greater than 214 + 1024 bytes.

• unknown_ca: A valid certificate chain or partial chain was received, but the certificate was not accepted because the CA certificate could not be located or could not be matched with a known, trusted CA.

• access_denied: A valid certificate was received, but when access control was applied, the sender decided not to proceed with the negotiation.

• decode_error: A message could not be decoded, because either a field was out of its specified range or the length of the message was incorrect.

• protocol_version: The protocol version the client attempted to negoti-ate is recognized but not supported.

• insufficient_security: Returned instead of handshake_failure when a negotiation has failed specifically because the server requires ciphers more secure than those supported by the client.

• unsupported_extension: Sent by clients that receive an extended server hello containing an extension not in the corresponding client hello.

• internal_error: An internal error unrelated to the peer or the correct-ness of the protocol makes it impossible to continue.

• decrypt_error: A handshake cryptographic operation failed, including being unable to verify a signature, decrypt a key exchange, or validate a fin-ished message.

The remaining alerts include the following.

• user_canceled: This handshake is being canceled for some reason unre-lated to a protocol failure.

• no_renegotiation: Sent by a client in response to a hello request or by the serverinresponsetoaclienthelloafterinitialhandshaking.Eitherofthesemessages would normally result in renegotiation, but this alert indicates that the sender is not able to renegotiate. This message is always a warning.


Cipher Suites

There are several small differences between the cipher suites available under SSLv3 and under TLS:

• Key Exchange: TLS supports all of the key exchange techniques of SSLv3 with the exception of Fortezza.

• Symmetric Encryption Algorithms: TLS includes all of the symmetric encryp-tion algorithms found in SSLv3, with the exception of Fortezza.

Client Certificate Types

TLS defines the following certificate types to be requested in a certificate_ request message: rsa_sign, dss_sign, rsa_fixed_dh, and dss_fixed_dh. These are all defined in SSLv3. In addition, SSLv3 includes rsa_ephemeral_dh, dss_ephemeral_dh, and fortezza_kea.EphemeralDiffie-HellmaninvolvessigningtheDiffie-HellmanparameterswitheitherRSAorDSS.ForTLS,thersa_sign and dss_sign types are used for that function; a separate signing type is notneededtosignDiffie-Hellmanparameters.TLSdoesnotincludetheFortezzascheme.

certificate_verify and Finished Messages

In the TLS certificate_verifymessage,theMD5andSHA-1hashesarecal-culated only over handshake_messages. Recall that for SSLv3, the hash calcula-tion also included the master secret and pads. These extra fields were felt to add no additional security.

As with the finished message in SSLv3, the finished message in TLS is a hash based on the shared master_secret, the previous handshake messages, and a label that identifies client or server. The calculation is somewhat different. For TLS, we have

PRF( master_secret,finished_label,MD5(handshake_messages)} SHA-1(handshake_messages))

where finished_label is the string “client finished” for the client and “server finished” for the server.

Cryptographic Computations

The pre_master_secret for TLS is calculated in the same way as in SSLv3. As in SSLv3, the master_secret in TLS is calculated as a hash function of the pre_master_secret and the two hello random numbers. The form of the TLS calcula-tion is different from that of SSLv3 and is defined as

master_secret = PRF(pre_master_secret,"master secret", ClientHello.random }ServerHello.random)

The algorithm is performed until 48 bytes of pseudorandom output are produced.

17.4 / httpS 543

The calculation of the key block material (MAC secret keys, session encryption keys, and IVs) is defined as

key_block = PRF(master_secret, "key expansion", SecurityParameters.server_random } SecurityParameters.client_random)

until enough output has been generated. As with SSLv3, the key_block is a func-tion of the master_secret and the client and server random numbers, but for TLS, the actual algorithm is different.

Padding

In SSL, the padding added prior to encryption of user data is the minimum amount required so that the total size of the data to be encrypted is a multiple of the cipher’s block length. In TLS, the padding can be any amount that results in a total that is a multiple of the cipher’s block length, up to a maximum of 255 bytes. For example, if the plaintext (or compressed text if compression is used) plus MAC plus padding.length byte is 79 bytes long, then the padding length (in bytes) can be 1, 9, 17, and so on, up to 249. A variable padding length may be used to frustrate attacks based on an analysis of the lengths of exchanged messages.

17.4 Https

HTTPS (HTTP over SSL) refers to the combination of HTTP and SSL to im-plement secure communication between a Web browser and a Web server. The HTTPS capability is built into all modern Web browsers. Its use depends on the Web server supporting HTTPS communication. For example, some search engines do not support HTTPS. Google provides HTTPS as an option: https://google.com.

The principal difference seen by a user of a Web browser is that URL (uni-form resource locator) addresses begin with https:// rather than http://. A normal HTTP connection uses port 80. If HTTPS is specified, port 443 is used, which invokes SSL.

When HTTPS is used, the following elements of the communication are encrypted:

• URL of the requested document

• Contents of the document

• Contents of browser forms (filled in by browser user)

• Cookies sent from browser to server and from server to browser

• Contents of HTTP header

HTTPS is documented in RFC 2818, HTTP Over TLS. There is no fundamen-tal change in using HTTP over either SSL or TLS, and both implementations are referred to as HTTPS.

https://google.com


Connection Initiation

For HTTPS, the agent acting as the HTTP client also acts as the TLS client. The client initiates a connection to the server on the appropriate port and then sends the TLS ClientHello to begin the TLS handshake. When the TLS handshake has finished, the client may then initiate the first HTTP request. All HTTP data is to be sent as TLS application data. Normal HTTP behavior, including retained connec-tions, should be followed.

There are three levels of awareness of a connection in HTTPS. At the HTTP level, an HTTP client requests a connection to an HTTP server by sending a con-nection request to the next lowest layer. Typically, the next lowest layer is TCP, but it also may be TLS/SSL. At the level of TLS, a session is established between a TLS client and a TLS server. This session can support one or more connections at any time. As we have seen, a TLS request to establish a connection begins with the establishment of a TCP connection between the TCP entity on the client side and the TCP entity on the server side.

Connection Closure

An HTTP client or server can indicate the closing of a connection by including the following line in an HTTP record: Connection: close. This indicates that the connection will be closed after this record is delivered.

The closure of an HTTPS connection requires that TLS close the connection with the peer TLS entity on the remote side, which will involve closing the underly-ing TCP connection. At the TLS level, the proper way to close a connection is for each side to use the TLS alert protocol to send a close_notify alert. TLS imple-mentations must initiate an exchange of closure alerts before closing a connection. A TLS implementation may, after sending a closure alert, close the connection without waiting for the peer to send its closure alert, generating an “incomplete close”. Note that an implementation that does this may choose to reuse the ses-sion. This should only be done when the application knows (typically through de-tecting HTTP message boundaries) that it has received all the message data that it cares about.

HTTP clients also must be able to cope with a situation in which the under-lying TCP connection is terminated without a prior close_notify alert and without a Connection: close indicator. Such a situation could be due to a programming error on the server or a communication error that causes the TCP connection to drop. However, the unannounced TCP closure could be evidence of some sort of attack. So the HTTPS client should issue some sort of security warning when this occurs.

17.5 secure sHeLL (ssH)

Secure Shell (SSH) is a protocol for secure network communications designed to be relatively simple and inexpensive to implement. The initial version, SSH1 wasfocusedonprovidingasecureremotelogonfacilitytoreplaceTELNETandother remote logon schemes that provided no security. SSH also provides a more

17.5 / Secure SheLL (SSh) 545

general client/server capability and can be used for such network functions as file transfer and e-mail. A new version, SSH2, fixes a number of security flaws in the originalscheme.SSH2isdocumentedasaproposedstandardinIETFRFCs4250through 4256.

SSH client and server applications are widely available for most operating sys-tems. It has become the method of choice for remote login and X tunneling and is rapidly becoming one of the most pervasive applications for encryption technology outside of embedded systems.

SSH is organized as three protocols that typically run on top of TCP (Figure 17.8):

• Transport Layer Protocol: Provides server authentication, data confidentiality, and data integrity with forward secrecy (i.e., if a key is compromised during one session, the knowledge does not affect the security of earlier sessions). The transport layer may optionally provide compression.

• User Authentication Protocol: Authenticates the user to the server.

• Connection Protocol: Multiplexes multiple logical communications channels over a single, underlying SSH connection.

Transport Layer Protocol

host Keys Server authentication occurs at the transport layer, based on the server possessing a public/private key pair. A server may have multiple host keys using multiple different asymmetric encryption algorithms. Multiple hosts may share the same host key. In any case, the server host key is used during key exchange to au-thenticate the identity of the host. For this to be possible, the client must have a

SSH UserAuthentication Protocol

SSH Transport Layer Protocol

TCP

IPInternet protocol provides datagram delivery acrossmultiple networks.

Transmission control protocol provides reliable, connection-oriented end-to-end delivery.

Provides server authentication, confidentiality, and integrity.It may optionally also provide compression.

Authenticates the client-sideuser to the server.

SSHConnection Protocol

Multiplexes the encryptedtunnel into several logicalchannels.

Figure 17.8 SSH Protocol Stack


priori knowledge of the server’s public host key. RFC 4251 dictates two alternative trust models that can be used:

1. The client has a local database that associates each host name (as typed by the user) with the corresponding public host key. This method requires no centrally administered infrastructure and no third-party coordination. The downside is that the database of name-to-key associations may become bur-densome to maintain.

2. The host name-to-key association is certified by a trusted certification author-ity (CA). The client only knows the CA root key and can verify the validity of all host keys certified by accepted CAs. This alternative eases the mainte-nance problem, since ideally, only a single CA key needs to be securely stored on the client. On the other hand, each host key must be appropriately certified by a central authority before authorization is possible.

PacKet exchange Figure 17.9 illustrates the sequence of events in the SSH Transport Layer Protocol. First, the client establishes a TCP connection to the server. This is done via the TCP protocol and is not part of the Transport Layer Protocol. Once the connection is established, the client and server exchange data,

Client Server

SSH-protoversion-softwareversionIdentification string

exchange

Algorithmnegotiation

End ofkey exchange

Servicerequest

SSH-protoversion-softwareversion

SSH_MSG_KEXINIT

SSH_MSG_KEXINIT

SSH_MSG_NEWKEYS

SSH_MSG_NEWKEYS

SSH_MSG_SERVICE_REQUEST

Establish TCP Connection

Key Exchange

Figure 17.9 SSHTransportLayerProtocolPacketExchanges


referredtoaspackets,inthedatafieldofaTCPsegment.Eachpacketisinthefol-lowing format (Figure 17.10).

• Packet length: Length of the packet in bytes, not including the packet length and MAC fields.

• Padding length: Length of the random padding field.

• Payload: Useful contents of the packet. Prior to algorithm negotiation, this field is uncompressed. If compression is negotiated, then in subsequent pack-ets, this field is compressed.

• Random padding: Once an encryption algorithm has been negotiated, this field is added. It contains random bytes of padding so that that total length of the packet (excluding the MAC field) is a multiple of the cipher block size, or 8 bytes for a stream cipher.

• Message authentication code (MAC): If message authentication has been ne-gotiated, this field contains the MAC value. The MAC value is computed over the entire packet plus a sequence number, excluding the MAC field. The se-quence number is an implicit 32-bit packet sequence that is initialized to zero for the first packet and incremented for every packet. The sequence number is not included in the packet sent over the TCP connection.

pdlpktl

pktl = packet lengthpdl = padding length

gniddaP# qes

Payload

SSH Packet

Compressed payload

Ciphertext

COMPRESS

ENCRYPT MAC

Figure 17.10 SSH Transport Layer Protocol Packet Formation


Once an encryption algorithm has been negotiated, the entire packet (exclud-ing the MAC field) is encrypted after the MAC value is calculated.

The SSH Transport Layer packet exchange consists of a sequence of steps (Figure 17.9). The first step, the identification string exchange, begins with the cli-ent sending a packet with an identification string of the form:

SSH-protoversion-softwareversion SP comments CR LF

where SP, CR, and LF are space character, carriage return, and line feed, respec-tively. An example of a valid string is SSH-2.0-billsSSH_3.6.3q3<CR><LF>. The server responds with its own identification string. These strings are used in the Diffie-Hellmankeyexchange.

Next comes algorithm negotiation.EachsidesendsanSSH_MSG_KEXINIT containing lists of supported algorithms in the order of preference to the sender. There is one list for each type of cryptographic algorithm. The algorithms include key exchange, encryption, MAC algorithm, and compression algorithm. Table 17.3 shows the allowable options for encryption, MAC, and compression. For each cat-egory, the algorithm chosen is the first algorithm on the client’s list that is also sup-ported by the server.

Table 17.3 SSH Transport Layer Cryptographic Algorithms

Cipher MAC algorithm

3des-cbc* Three-key3DESin CBC mode

hmac-sha1* HMAC-SHA1; digest length = key length = 20

blowfish-cbc Blowfish in CBC mode hmac-sha1-96** First 96 bits of HMAC-SHA1; digest length = 12; key length = 20

twofish256-cbc Twofish in CBC mode with a 256-bit key

hmac-md5 HMAC-MD5;digest length = key length = 16

twofish192-cbc Twofish with a 192-bit key hmac-md5-96 First96bitsofHMAC-MD5;digest length = 12; key length = 16

twofish128-cbc Twofish with a 128-bit key

aes256-cbc AESinCBCmodewitha256-bit key Compression algorithm

aes192-cbc AESwitha192-bitkey none* No compression

aes128-cbc** AESwitha128-bitkey zlib DefinedinRFC1950and RFC 1951

Serpent256-cbc Serpent in CBC mode with a 256-bit key

Serpent192-cbc Serpent with a 192-bit key

Serpent128-cbc Serpent with a 128-bit key

arcfour RC4 with a 128-bit key

cast128-cbc CAST-128 in CBC mode

* = Required** = Recommended


The next step is key exchange. The specification allows for alternative methodsofkeyexchange,butatpresent,onlytwoversionsofDiffie-Hellmankeyexchange are specified. Both versions are defined in RFC 2409 and require only one packet in each direction. The following steps are involved in the exchange. In this, C is the client; S is the server; p is a large safe prime; g is a generator for a subgroup of GF(p); q is the order of the subgroup; V_S is S’s identification string; V_C is C’s identification string; K_S is S’s public host key; I_C is C’s SSH_MSG_KEXINIT message and I_S is S’s SSH_MSG_KEXINIT message that have been exchanged before this part begins. The values of p, g, and q are known to both client and server as a result of the algorithm selection negotiation. The hash function hash() is also decided during algorithm negotiation.

1. C generates a random number x(1 6 x 6 q) and computes e = gx mod p. C sends e to S.

2. S generates a random number y(0 6 y 6 q) and computes f = gy mod p. S receives e. It computes K = ey mod p, H = hash(V_C }V_S } I_C } I_S }K_S } e } f }K), and signature s on H with its private host key. S sends (K_S } f } s) to C. The signing operation may involve a second hashing operation.

3. C verifies that K_S really is the host key for S (e.g., using certificates or a local database). C is also allowed to accept the key without verification; however, doing so will render the protocol insecure against active attacks (but may be desirable for practical reasons in the short term in many environments). C then computes K = f x mod p, H = hash(V_C }V_S } I_C } I_S }K_S } e } f }K), and verifies the signature s on H.

As a result of these steps, the two sides now share a master key K. In addition, the server has been authenticated to the client, because the server has used its pri-vatekeytosignitshalfoftheDiffie-Hellmanexchange.Finally,thehashvalueH serves as a session identifier for this connection. Once computed, the session identi-fier is not changed, even if the key exchange is performed again for this connection to obtain fresh keys.

The end of key exchange is signaled by the exchange of SSH_MSG_NEWKEYS packets. At this point, both sides may start using the keys generated from K, as discussed subsequently.

The final step is service request. The client sends an SSH_MSG_SERVICE_REQUEST packet to request either the User Authentication or the Connection Protocol. Subsequent to this, all data is exchanged as the payload of an SSH Transport Layer packet, protected by encryption and MAC.

Key generation The keys used for encryption and MAC (and any needed IVs) are generated from the shared secret key K, the hash value from the key exchange H, and the session identifier, which is equal to H unless there has been a subsequent key exchange after the initial key exchange. The values are computed as follows.

• Initial IV client to server: HASH(K }H }"A" } session_id)

• Initial IV server to client: HASH(K }H }"B" } session_id)

• Encryptionkeyclienttoserver:HASH(K }H }"C" } session_id)


• Encryptionkeyservertoclient:HASH(K }H }"D" } session_id)

• Integrity key client to server: HASH(K }H }"E" } session_id)

• Integrity key server to client: HASH(K }H }"F" } session_id)

where HASH() is the hash function determined during algorithm negotiation.

User Authentication Protocol

The User Authentication Protocol provides the means by which the client is au-thenticated to the server.

Message tyPes and ForMats Three types of messages are always used in the User Authentication Protocol. Authentication requests from the client have the format:

byte SSH_MSG_USERAUTH_REQUEST (50)

string user namestring service namestring method name… method specific fields

where user name is the authorization identity the client is claiming, service name is the facility to which the client is requesting access (typically the SSH Connection Protocol), and method name is the authentication method being used in this request. The first byte has decimal value 50, which is interpreted as SSH_MSG_USERAUTH_REQUEST.

Iftheservereither(1)rejectstheauthenticationrequestor(2)acceptsthere-quest but requires one or more additional authentication methods, the server sends a message with the format:

byte SSH_MSG_USERAUTH_FAILURE (51)

name-list authentications that can continueboolean partial success

where the name-list is a list of methods that may productively continue the dia-log. If the server accepts authentication, it sends a single byte message: SSH_MSG_ USERAUTH_SUCCESS (52).

Message exchange The message exchange involves the following steps.

1. The client sends a SSH_MSG_USERAUTH_REQUEST with a requested method of none.

2. The server checks to determine if the user name is valid. If not, the server re-turns SSH_MSG_USERAUTH_FAILURE with the partial success value of false. If the user name is valid, the server proceeds to step 3.

3. The server returns SSH_MSG_USERAUTH_FAILURE with a list of one or more authentication methods to be used.

4. The client selects one of the acceptable authentication methods and sends a SSH_MSG_USERAUTH_REQUEST with that method name and the required


method-specific fields. At this point, there may be a sequence of exchanges to perform the method.

5. If the authentication succeeds and more authentication methods are required, the server proceeds to step 3, using a partial success value of true. If the au-thentication fails, the server proceeds to step 3, using a partial success value of false.

6. When all required authentication methods succeed, the server sends a SSH_MSG_USERAUTH_SUCCESS message, and the Authentication Protocol is over.

authentication Methods The server may require one or more of the following authentication methods.

• publickey: The details of this method depend on the public-key algorithm chosen. In essence, the client sends a message to the server that contains the client’s public key, with the message signed by the client’s private key. When the server receives this message, it checks whether the supplied key is accept-able for authentication and, if so, it checks whether the signature is correct.

• password: The client sends a message containing a plaintext password, which is protected by encryption by the Transport Layer Protocol.

• hostbased: Authentication is performed on the client’s host rather than the client itself. Thus, a host that supports multiple clients would provide authen-tication for all its clients. This method works by having the client send a signa-ture created with the private key of the client host. Thus, rather than directly verifying the user’s identity, the SSH server verifies the identity of the client host—and then believes the host when it says the user has already authenti-cated on the client side.

Connection Protocol

The SSH Connection Protocol runs on top of the SSH Transport Layer Protocol and assumes that a secure authentication connection is in use.2 That secure authen-tication connection, referred to as a tunnel, is used by the Connection Protocol to multiplex a number of logical channels.

channel MechanisM All types of communication using SSH, such as a terminal session,aresupportedusingseparatechannels.Eithersidemayopenachannel.Foreach channel, each side associates a unique channel number, which need not be the same on both ends. Channels are flow controlled using a window mechanism. No data may be sent to a channel until a message is received to indicate that window space is available.

2RFC 4254, The Secure Shell (SSH) Connection Protocol, states that the Connection Protocol runs on top of the Transport Layer Protocol and the User Authentication Protocol. RFC 4251, SSH Protocol Architecture, states that the Connection Protocol runs over the User Authentication Protocol. In fact, the Connection Protocol runs over the Transport Layer Protocol, but assumes that the User Authentication Protocol has been previously invoked.


The life of a channel progresses through three stages: opening a channel, data transfer, and closing a channel.

When either side wishes to open a new channel, it allocates a local number for the channel and then sends a message of the form:

byte SSH_MSG_CHANNEL_OPEN

string channel typeuint32 sender channeluint32 initial window sizeuint32 maximum packet size…. channel type specific data follows

where uint32 means unsigned 32-bit integer. The channel type identifies the appli-cation for this channel, as described subsequently. The sender channel is the local channel number. The initial window size specifies how many bytes of channel data canbesenttothesenderofthismessagewithoutadjustingthewindow.Themaxi-mum packet size specifies the maximum size of an individual data packet that can be sent to the sender. For example, one might want to use smaller packets for inter-active connections to get better interactive response on slow links.

If the remote side is able to open the channel, it returns a SSH_MSG_ CHANNEL_OPEN_CONFIRMATION message, which includes the sender channel number, the recipient channel number, and window and packet size values for in-coming traffic. Otherwise, the remote side returns a SSH_MSG_CHANNEL_OPEN_FAILURE message with a reason code indicating the reason for failure.

Once a channel is open, data transfer is performed using a SSH_MSG_ CHANNEL_DATA message, which includes the recipient channel number and a block of data. These messages, in both directions, may continue as long as the channel isopen.

When either side wishes to close a channel, it sends a SSH_MSG_CHANNEL_CLOSE message, which includes the recipient channel number.

Figure 17.11 provides an example of Connection Protocol Message Exchange.

channel tyPes Four channel types are recognized in the SSH Connection Protocol specification.

• session: The remote execution of a program. The program may be a shell, an application such as file transfer or e-mail, a system command, or some built-in subsystem. Once a session channel is opened, subsequent requests are used to start the remote program.

• x11: This refers to the X Window System, a computer software system and network protocol that provides a graphical user interface (GUI) for net-worked computers. X allows applications to run on a network server but to be displayed on a desktop machine.

• forwarded-tcpip: This is remote port forwarding, as explained in the next subsection.

• direct-tcpip: This is local port forwarding, as explained in the next subsection.


Port Forwarding One of the most useful features of SSH is port forwarding. In es-sence, port forwarding provides the ability to convert any insecure TCP connection into a secure SSH connection. This is also referred to as SSH tunneling. We need to know what a port is in this context. A port is an identifier of a user of TCP. So, any application that runs on top of TCP has a port number. Incoming TCP traffic is de-livered to the appropriate application on the basis of the port number. An applica-tion may employ multiple port numbers. For example, for the Simple Mail Transfer Protocol (SMTP), the server side generally listens on port 25, so an incoming SMTP request uses TCP and addresses the data to destination port 25. TCP recognizes that this is the SMTP server address and routes the data to the SMTP server application.

Figure 17.12 illustrates the basic concept behind port forwarding. We have a client application that is identified by port number x and a server application identi-fied by port number y. At some point, the client application invokes the local TCP entity and requests a connection to the remote server on port y. The local TCP

Client Server

SSH_MSG_CHANNEL_OPENOpen achannel

Datatransfer

Close achannel

SSH_MSG_CHANNEL_OPEN_CONFIRMATION

SSH_MSG_CHANNEL_DATA




SSH_MSG_CHANNEL_CLOSE

Establish Authenticated Transport Layer Connection

Figure 17.11 ExampleofSSHConnectionProtocolMessageExchange


entity negotiates a TCP connection with the remote TCP entity, such that the con-nection links local port x to remote port y.

To secure this connection, SSH is configured so that the SSH Transport Layer Protocol establishes a TCP connection between the SSH client and server entities, with TCP port numbers a and b, respectively. A secure SSH tunnel is established over this TCP connection. Traffic from the client at port x is redirected to the local SSH entity and travels through the tunnel where the remote SSH entity delivers the data to the server application on port y. Traffic in the other direction is similarly redirected.

SSH supports two types of port forwarding: local forwarding and remote for-warding. Local forwardingallowstheclienttosetupa“hijacker”process.Thiswillintercept selected application-level traffic and redirect it from an unsecured TCP connection to a secure SSH tunnel. SSH is configured to listen on selected ports. SSH grabs all traffic using a selected port and sends it through an SSH tunnel. On the other end, the SSH server sends the incoming traffic to the destination port dic-tated by the client application.

ClientServer

Clientapplication

Unsecure TCP connection

(a) Connection via TCP

TCPentity

x y

Serverapplication

TCPentity

Clientapplication

Secure SSH tunnel

(b) Connection via SSH tunnel

SSHentity

x y

Serverapplication

SSHentity

Unsecure TCP connectionTCPentity

a b

TCPentity

Figure 17.12 SSHTransportLayerPacketExchanges

17.6 / recommended reading 555

The following example should help clarify local forwarding. Suppose you have an e-mail client on your desktop and use it to get e-mail from your mail server via the Post Office Protocol (POP). The assigned port number for POP3 is port 110. We can secure this traffic in the following way:

1. The SSH client sets up a connection to the remote server.

2. Select an unused local port number, say 9999, and configure SSH to accept traffic from this port destined for port 110 on the server.

3. The SSH client informs the SSH server to create a connection to the destina-tion, in this case mailserver port 110.

4. The client takes any bits sent to local port 9999 and sends them to the server inside the encrypted SSH session. The SSH server decrypts the incoming bits and sends the plaintext to port 110.

5. In the other direction, the SSH server takes any bits received on port 110 and sends them inside the SSH session back to the client, who decrypts and sends them to the process connected to port 9999.

With remote forwarding, the user’s SSH client acts on the server’s behalf. The client receives traffic with a given destination port number, places the traffic on the correct port and sends it to the destination the user chooses. A typical example of remote forwarding is the following. You wish to access a server at work from your home computer. Because the work server is behind a firewall, it will not accept an SSH request from your home computer. However, from work you can set up an SSH tunnel using remote forwarding. This involves the following steps.

1. From the work computer, set up an SSH connection to your home computer. The firewall will allow this, because it is a protected outgoing connection.

2. Configure the SSH server to listen on a local port, say 22, and to deliver data across the SSH connection addressed to remote port, say 2222.

3. You can now go to your home computer, and configure SSH to accept traffic on port 2222.

4. You now have an SSH tunnel that can be used for remote logon to the work server.


[RESC01]isagooddetailedtreatmentofSSLandTLS.[BARR05]providesathoroughtreatmentofSSH.Theoriginalversion(SSH-1)ofSSHwasintroducedin[YLON96].

BARR05 Barrett,D.;Silverman,R.;andByrnes,R.SSH The Secure Shell: The Definitive Guide. Sebastopol, CA: O’Reilly, 2005.

RESC01 Rescorla,E.SSL and TLS: Designing and Building Secure Systems. Reading, MA: Addison-Wesley, 2001.

YLON96 Ylonen, T. “SSH - Secure Login Connections over the Internet.” Proceedings, Sixth USENIX Security Symposium, July 1996.


17.7 key terms, revieW QuestiOns, and prObLems

Key Terms

Alert protocolChange Cipher Spec protocolHandshake protocol

HTTPS (HTTP over SSL)Master SecretSecure Shell (SSH)

Secure Socket Layer (SSL)Transport Layer Security

(TLS)

Review Questions 17.1 What are the advantages of each of the three approaches shown in Figure 17.1? 17.2 What protocols comprise SSL? 17.3 What is the difference between an SSL connection and an SSL session? 17.4 List and briefly define the parameters that define an SSL session state. 17.5 List and briefly define the parameters that define an SSL session connection. 17.6 What services are provided by the SSL Record Protocol? 17.7 What steps are involved in the SSL Record Protocol transmission? 17.8 What is the purpose of HTTPS? 17.9 For what applications is SSH useful? 17.10 List and briefly define the SSH protocols.

Problems 17.1 In SSL and TLS, why is there a separate Change Cipher Spec Protocol rather than

including a change_cipher_spec message in the Handshake Protocol? 17.2 What purpose does the MAC serve during the change cipher spec SSL exchange? 17.3 Consider the following threats to Web security and describe how each is countered by

a particular feature of SSL. a. Brute-Force Cryptanalytic Attack: An exhaustive search of the key space for a

conventional encryption algorithm. b. Known Plaintext Dictionary Attack: Many messages will contain predictable

plaintext, such as theHTTPGET command.An attacker constructs a diction-ary containing every possible encryption of the known-plaintext message. When an encrypted message is intercepted, the attacker takes the portion containing the encrypted known plaintext and looks up the ciphertext in the dictionary. The ciphertext should match against an entry that was encrypted with the same secret key. If there are several matches, each of these can be tried against the full cipher-text to determine the right one. This attack is especially effective against small key sizes (e.g., 40-bit keys).

c. ReplayAttack:EarlierSSLhandshakemessagesarereplayed. d. Man-in-the-Middle Attack: An attacker interposes during key exchange, acting as

the client to the server and as the server to the client. e. Password Sniffing: Passwords in HTTP or other application traffic are eaves-

dropped. f. IP Spoofing: Uses forged IP addresses to fool a host into accepting bogus data. g. IPHijacking:Anactive,authenticatedconnectionbetweentwohostsisdisrupted

and the attacker takes the place of one of the hosts.

17.7 / key termS, revieW QueStionS, and probLemS 557

h. SYN Flooding: An attacker sends TCP SYN messages to request a connection but does not respond to the final message to establish the connection fully. The at-tacked TCP module typically leaves the “half-open connection” around for a few minutes. Repeated SYN messages can clog the TCP module.

17.4 Based on what you have learned in this chapter, is it possible in SSL for the receiver to reorder SSL record blocks that arrive out of order? If so, explain how it can be done. If not, why not?

17.5 For SSH packets, what is the advantage, if any, of not including the MAC in the scope of the packet encryption?

Wireless NetWork security18.1 Wireless Security

Wireless Network Threats Wireless Security Measures

18.2 Mobile Device Security

Security Threats Mobile Device Security Strategy

18.3 IEEE 802.11 Wireless LAN Overview

The Wi-Fi Alliance IEEE 802 Protocol Architecture IEEE 802.11 Network Components and Architectural Model IEEE 802.11 Services

18.4 IEEE 802.11i Wireless LAN Security

IEEE 802.11i Services IEEE 802.11i Phases of Operation Discovery Phase Authentication Phase Key Management Phase Protected Data Transfer Phase The IEEE 802.11i Pseudorandom Function



Chapter

558558

18.1 / Wireless security 559

Learning Objectives


u Present an overview of security threats and countermeasures for wireless networks.

u Understand the unique security threats posed by the use of mobile devices with enterprise networks.

u Describe the principal elements in a mobile device security strategy.

u Understand the essential elements of the IEEE 802.11 wireless LAN standard.

u Summarize the various components of the IEEE 802.11i wireless LAN security architecture.

This chapter begins with a general overview of wireless security issues. We then focus on the relatively new area of mobile device security, examining threats and counter-measures for mobile devices used in the enterprise. Then, we look at the IEEE 802.11i standard for wireless LAN security. This standard is part of IEEE 802.11, also referred to as Wi-Fi. We begin the discussion with an overview of IEEE 802.11, and then we look in some detail at IEEE 802.11i.

18.1 WireLess security

Wireless networks, and the wireless devices that use them, introduce a host of secu-rity problems over and above those found in wired networks. Some of the key fac-tors contributing to the higher security risk of wireless networks compared to wired networks include the following [MA10]:

• Channel: Wireless networking typically involves broadcast communications, which is far more susceptible to eavesdropping and jamming than wired net-works. Wireless networks are also more vulnerable to active attacks that ex-ploit vulnerabilities in communications protocols.

Investigators have published numerous reports of birds taking turns vocalizing; the bird spoken to gave its full attention to the speaker and never vocalized at the same time, as if the two were holding a conversation.

Researchers and scholars who have studied the data on avian communication care-fully write (a) the communication code of birds, such as crows, has not been broken by any means; (b) probably all birds have wider vocabularies than anyone realizes; and (c) greater complexity and depth are recognized in avian communication as research progresses.

—The Human Nature of Birds, Theodore Barber

560 chapter 18 / Wireless NetWork security

• Mobility: Wireless devices are, in principal and usually in practice, far more portable and mobile than wired devices. This mobility results in a number of risks, described subsequently.

• Resources: Some wireless devices, such as smartphones and tablets, have so-phisticated operating systems but limited memory and processing resources with which to counter threats, including denial of service and malware.

• Accessibility: Some wireless devices, such as sensors and robots, may be left unattended in remote and/or hostile locations. This greatly increases their vul-nerability to physical attacks.

In simple terms, the wireless environment consists of three components that provide point of attack (Figure 18.1). The wireless client can be a cell phone, a Wi-Fi–enabled laptop or tablet, a wireless sensor, a Bluetooth device, and so on. The wireless access point provides a connection to the network or service. Examples of access points are cell towers, Wi-Fi hotspots, and wireless access points to wired local or wide area networks. The transmission medium, which carries the radio waves for data transfer, is also a source of vulnerability.

Wireless Network Threats

[CHOI08] lists the following security threats to wireless networks:

• Accidental association: Company wireless LANs or wireless access points to wired LANs in close proximity (e.g., in the same or neighboring buildings) may create overlapping transmission ranges. A user intending to connect to one LAN may unintentionally lock on to a wireless access point from a neigh-boring network. Although the security breach is accidental, it nevertheless ex-poses resources of one LAN to the accidental user.

• Malicious association: In this situation, a wireless device is configured to ap-pear to be a legitimate access point, enabling the operator to steal passwords from legitimate users and then penetrate a wired network through a legitimate wireless access point.

• Ad hoc networks: These are peer-to-peer networks between wireless comput-ers with no access point between them. Such networks can pose a security threat due to a lack of a central point of control.

• Nontraditional networks: Nontraditional networks and links, such as personal network Bluetooth devices, barcode readers, and handheld PDAs, pose a se-curity risk in terms of both eavesdropping and spoofing.

Endpoint Wireless medium Access point

Figure 18.1 Wireless Networking Components

18.1 / Wireless security 561

• Identity theft (MAC spoofing): This occurs when an attacker is able to eaves-drop on network traffic and identify the MAC address of a computer with network privileges.

• Man-in-the middle attacks: This type of attack is described in Chapter 10 in the context of the Diffie-Hellman key exchange protocol. In a broader sense, this attack involves persuading a user and an access point to believe that they are talking to each other when in fact the communication is going through an intermediate attacking device. Wireless networks are particularly vulnerable to such attacks.

• Denial of service (DoS): This type of attack is discussed in detail in Chapter 21. In the context of a wireless network, a DoS attack occurs when an attacker continually bombards a wireless access point or some other accessible wireless port with various protocol messages designed to consume system resources. The wireless environment lends itself to this type of attack, because it is so easy for the attacker to direct multiple wireless messages at the target.

• Network injection: A network injection attack targets wireless access points that are exposed to nonfiltered network traffic, such as routing protocol mes-sages or network management messages. An example of such an attack is one in which bogus reconfiguration commands are used to affect routers and switches to degrade network performance.

Wireless Security Measures

Following [CHOI08], we can group wireless security measures into those dealing with wireless transmissions, wireless access points, and wireless networks (consist-ing of wireless routers and endpoints).

Securing WireleSS TranSmiSSionS The principal threats to wireless transmission are eavesdropping, altering or inserting messages, and disruption. To deal with eavesdropping, two types of countermeasures are appropriate:

• Signal-hiding techniques: Organizations can take a number of measures to make it more difficult for an attacker to locate their wireless access points, including turning off service set identifier (SSID) broadcasting by wireless ac-cess points; assigning cryptic names to SSIDs; reducing signal strength to the lowest level that still provides requisite coverage; and locating wireless access points in the interior of the building, away from windows and exterior walls. Greater security can be achieved by the use of directional antennas and of signal-shielding techniques.

• Encryption: Encryption of all wireless transmission is effective against eaves-dropping to the extent that the encryption keys are secured.

The use of encryption and authentication protocols is the standard method of countering attempts to alter or insert transmissions.

The methods discussed in Chapter 21 for dealing with DoS apply to wireless transmissions. Organizations can also reduce the risk of unintentional DoS attacks. Site surveys can detect the existence of other devices using the same frequency


range, to help determine where to locate wireless access points. Signal strengths can be adjusted and shielding used in an attempt to isolate a wireless environment from competing nearby transmissions.

Securing WireleSS acceSS PoinTS The main threat involving wireless access points is unauthorized access to the network. The principal approach for preventing such access is the IEEE 802.1X standard for port-based network access control. The standard provides an authentication mechanism for devices wishing to attach to a LAN or wireless network. The use of 802.1X can prevent rogue access points and other unauthorized devices from becoming insecure backdoors.

Section 16.3 provides an introduction to 802.1X.

Securing WireleSS neTWorkS [CHOI08] recommends the following techniques for wireless network security:

1. Use encryption. Wireless routers are typically equipped with built-in encryp-tion mechanisms for router-to-router traffic.

2. Use antivirus and antispyware software, and a firewall. These facilities should be enabled on all wireless network endpoints.

3. Turn off identifier broadcasting. Wireless routers are typically configured to broadcast an identifying signal so that any device within range can learn of the router’s existence. If a network is configured so that authorized devices know the identity of routers, this capability can be disabled, so as to thwart attackers.

4. Change the identifier on your router from the default. Again, this measure thwarts attackers who will attempt to gain access to a wireless network using default router identifiers.

5. Change your router’s pre-set password for administration. This is another pru-dent step.

6. Allow only specific computers to access your wireless network. A router can be configured to only communicate with approved MAC addresses. Of course, MAC addresses can be spoofed, so this is just one element of a security strategy.

18.2 MObiLe Device security

Prior to the widespread use of smartphones, the dominant paradigm for computer and network security in organizations was as follows. Corporate IT was tightly con-trolled. User devices were typically limited to Windows PCs. Business applications were controlled by IT and either run locally on endpoints or on physical servers in data centers. Network security was based upon clearly defined perimeters that separated trusted internal networks from the untrusted Internet. Today, there have been massive changes in each of these assumptions. An organization’s networks must accommodate the following:

• Growing use of new devices: Organizations are experiencing significant growth in employee use of mobile devices. In many cases, employees are allowed to use a combination of endpoint devices as part of their day-to-day activities.

18.2 / Mobile Device security 563

• Cloud-based applications: Applications no longer run solely on physical servers in corporate data centers. Quite the opposite, applications can run anywhere—on traditional physical servers, on mobile virtual servers, or in the cloud. Additionally, end users can now take advantage of a wide variety of cloud-based applications and IT services for personal and professional use. Facebook can be used for an employee’s personal profiles or as a component of a corporate marketing campaign. Employees depend upon Skype to speak with friends abroad or for legitimate business video conferencing. Dropbox and Box can be used to distribute documents between corporate and personal devices for mobility and user productivity.

• De-perimeterization: Given new device proliferation, application mobility, and cloud-based consumer and corporate services, the notion of a static net-work perimeter is all but gone. Now there are a multitude of network perim-eters around devices, applications, users, and data. These perimeters have also become quite dynamic as they must adapt to various environmental conditions such as user role, device type, server virtualization mobility, network location, and time-of-day.

• External business requirements: The enterprise must also provide guests, third-party contractors, and business partners network access using various devices from a multitude of locations.

The central element in all of these changes is the mobile computing device. Mobile devices have become an essential element for organizations as part of the overall network infrastructure. Mobile devices such as smartphones, tablets, and memory sticks provide increased convenience for individuals as well as the potential for increased productivity in the workplace. Because of their widespread use and unique characteristics, security for mobile devices is a pressing and complex issue. In essence, an organization needs to implement a security policy through a combi-nation of security features built into the mobile devices and additional security con-trols provided by network components that regulate the use of the mobile devices.

Security Threats

Mobile devices need additional, specialized protection measures beyond those implemented for other client devices, such as desktop and laptop devices that are used only within the organization’s facilities and on the organization’s networks. SP 800-14 (Guidelines for Managing and Securing Mobile Devices in the Enterprise, July 2012) lists seven major security concerns for mobile devices. We examine each of these in turn.

lack of PhySical SecuriTy conTrolS Mobile devices are typically under the com-plete control of the user, and are used and kept in a variety of locations outside the organization’s control, including off premises. Even if a device is required to remain on premises, the user may move the device within the organization between secure and nonsecured locations. Thus, theft and tampering are realistic threats.

The security policy for mobile devices must be based on the assumption that any mobile device may be stolen or at least accessed by a malicious party. The threat


is twofold: A malicious party may attempt to recover sensitive data from the device itself, or may use the device to gain access to the organization’s resources.

uSe of unTruSTed mobile deviceS In addition to company-issued and company-controlled mobile devices, virtually all employees will have personal smartphones and/or tablets. The organization must assume that these devices are not trustwor-thy. That is, the devices may not employ encryption and either the user or a third party may have installed a bypass to the built-in restrictions on security, operating system use, and so on.

uSe of unTruSTed neTWorkS If a mobile device is used on premises, it can con-nect to organization resources over the organization’s own in-house wireless net-works. However, for off-premises use, the user will typically access organizational resources via Wi-Fi or cellular access to the Internet and from the Internet to the organization. Thus, traffic that includes an off-premises segment is potentially sus-ceptible to eavesdropping or man-in-the-middle types of attacks. Thus, the security policy must be based on the assumption that the networks between the mobile de-vice and the organization are not trustworthy.

uSe of aPPlicaTionS creaTed by unknoWn ParTieS By design, it is easy to find and install third-party applications on mobile devices. This poses the obvious risk of installing malicious software. An organization has several options for dealing with this threat, as described subsequently.

inTeracTion WiTh oTher SySTemS A common feature found on smartphones and tablets is the ability to automatically synchronize data, apps, contacts, photos, and so on with other computing devices and with cloud-based storage. Unless an orga-nization has control of all the devices involved in synchronization, there is consider-able risk of the organization’s data being stored in an unsecured location, plus the risk of the introduction of malware.

uSe of unTruSTed conTenT Mobile devices may access and use content that other computing devices do not encounter. An example is the Quick Response (QR) code, which is a two-dimensional barcode. QR codes are designed to be captured by a mobile device camera and used by the mobile device. The QR code translates to a URL, so that a malicious QR code could direct the mobile device to malicious Web sites.

uSe of locaTion ServiceS The GPS capability on mobile devices can be used to maintain a knowledge of the physical location of the device. While this feature might be useful to an organization as part of a presence service, it creates security risks. An attacker can use the location information to determine where the device and user are located, which may be of use to the attacker.

Mobile Device Security Strategy

With the threats listed in the preceding discussion in mind, we outline the principal elements of a mobile device security strategy. They fall into three categories: device security, client/server traffic security, and barrier security (Figure 18.2).

18.2 / Mobile Device security 565

device SecuriTy A number of organizations will supply mobile devices for em-ployee use and preconfigure those devices to conform to the enterprise security policy. However, many organizations will find it convenient or even necessary to adopt a bring-your-own-device (BYOD) policy that allows the personal mobile devices of employ-ees to have access to corporate resources. IT managers should be able to inspect each device before allowing network access. IT will want to establish configuration guide-lines for operating systems and applications. For example, “rooted” or “jail-broken” devices are not permitted on the network, and mobile devices cannot store corporate contacts on local storage. Whether a device is owned by the organization or BYOD, the organization should configure the device with security controls, including the following:

• Enable auto-lock, which causes the device to lock if it has not been used for a given amount of time, requiring the user to re-enter a four-digit PIN or a pass-word to re-activate the device.

• Enable password or PIN protection. The PIN or password is needed to unlock the device. In addition, it can be configured so that e-mail and other data on the device are encrypted using the PIN or password and can only be retrieved with the PIN or password.

• Avoid using auto-complete features that remember user names or passwords. • Enable remote wipe.

Firewall

Firewall limits scope of dataand applicationaccess

Authenticationand access controlprotocols used toverify device and userand establish limitson access

Mobile device isconfigured withsecurity mechanisms andparameters to conform toorganization security policy

Traffic is encrypted;uses SSL or IPsecVPN tunnel

Authentication/access controlserver

Mobile deviceconfigurationserver

Application/databaseserver

Figure 18.2 Mobile Device Security Elements


• Ensure that SSL protection is enabled, if available.

• Make sure that software, including operating systems and applications, is up to date.

• Install antivirus software as it becomes available.

• Either sensitive data should be prohibited from storage on the mobile device or it should be encrypted.

• IT staff should also have the ability to remotely access devices, wipe the device of all data, and then disable the device in the event of loss or theft.

• The organization may prohibit all installation of third-party applications, implement whitelisting to prohibit installation of all unapproved applica-tions, or implement a secure sandbox that isolates the organization’s data and applications from all other data and applications on the mobile device. Any application that is on an approved list should be accompanied by a digital signature and a public-key certificate from an approved authority.

• The organization can implement and enforce restrictions on what devices can synchronize and on the use of cloud-based storage.

• To deal with the threat of untrusted content, security responses can include training of personnel on the risks inherent in untrusted content and disabling camera use on corporate mobile devices.

• To counter the threat of malicious use of location services, the security policy can dictate that such service is disabled on all mobile devices.

Traffic SecuriTy Traffic security is based on the usual mechanisms for encryption and authentication. All traffic should be encrypted and travel by secure means, such as SSL or IPv6. Virtual private networks (VPNs) can be configured so that all traffic between the mobile device and the organization’s network is via a VPN.

A strong authentication protocol should be used to limit the access from the device to the resources of the organization. Often, a mobile device has a single device-specific authenticator, because it is assumed that the device has only one user. A preferable strategy is to have a two-layer authentication mechanism, which involves authenticating the device and then authenticating the user of the device.

barrier SecuriTy The organization should have security mechanisms to protect the network from unauthorized access. The security strategy can also include firewall pol-icies specific to mobile device traffic. Firewall policies can limit the scope of data and application access for all mobile devices. Similarly, intrusion detection and intrusion prevention systems can be configured to have tighter rules for mobile device traffic.

18.3 ieee 802.11 WireLess Lan OvervieW

IEEE 802 is a committee that has developed standards for a wide range of local area networks (LANs). In 1990, the IEEE 802 Committee formed a new work-ing group, IEEE 802.11, with a charter to develop a protocol and transmission specifications for wireless LANs (WLANs). Since that time, the demand for

18.3 / ieee 802.11 Wireless laN overvieW 567

WLANs at different frequencies and data rates has exploded. Keeping pace with this demand, the IEEE 802.11 working group has issued an ever-expanding list of standards. Table 18.1 briefly defines key terms used in the IEEE 802.11 standard.

The Wi-Fi Alliance

The first 802.11 standard to gain broad industry acceptance was 802.11b. Although 802.11b products are all based on the same standard, there is always a concern whether products from different vendors will successfully interoperate. To meet this concern, the Wireless Ethernet Compatibility Alliance (WECA), an indus-try consortium, was formed in 1999. This organization, subsequently renamed the Wi-Fi (Wireless Fidelity) Alliance, created a test suite to certify interoperability for 802.11b products. The term used for certified 802.11b products is Wi-Fi. Wi-Fi certi-fication has been extended to 802.11g products. The Wi-Fi Alliance has also devel-oped a certification process for 802.11a products, called Wi-Fi5. The Wi-Fi Alliance is concerned with a range of market areas for WLANs, including enterprise, home, and hot spots.

More recently, the Wi-Fi Alliance has developed certification procedures for IEEE 802.11 security standards, referred to as Wi-Fi Protected Access (WPA). The most recent version of WPA, known as WPA2, incorporates all of the features of the IEEE 802.11i WLAN security specification.

IEEE 802 Protocol Architecture

Before proceeding, we need to briefly preview the IEEE 802 protocol architecture. IEEE 802.11 standards are defined within the structure of a layered set of protocols. This structure, used for all IEEE 802 standards, is illustrated in Figure 18.3.

Table 18.1 IEEE 802.11 Terminology

Access point (AP) Any entity that has station functionality and provides access to the distribution system via the wireless medium for associated stations.

Basic service set (BSS) A set of stations controlled by a single coordination function.

Coordination function The logical function that determines when a station operating within a BSS is permitted to transmit and may be able to receive PDUs.

Distribution system (DS) A system used to interconnect a set of BSSs and integrated LANs to create an ESS.

Extended service set (ESS)

A set of one or more interconnected BSSs and integrated LANs that appear as a single BSS to the LLC layer at any station associated with one of these BSSs.

MAC protocol data unit (MPDU)

The unit of data exchanged between two peer MAC entities using the services of the physical layer.

MAC service data unit (MSDU)

Information that is delivered as a unit between MAC users.

Station Any device that contains an IEEE 802.11 conformant MAC and physical layer.


PhySical layer The lowest layer of the IEEE 802 reference model is the physical layer, which includes such functions as encoding/decoding of signals and bit trans-mission/reception. In addition, the physical layer includes a specification of the transmission medium. In the case of IEEE 802.11, the physical layer also defines frequency bands and antenna characteristics.

media acceSS conTrol All LANs consist of collections of devices that share the net-work’s transmission capacity. Some means of controlling access to the transmission medium is needed to provide an orderly and efficient use of that capacity. This is the function of a media access control (MAC) layer. The MAC layer receives data from a higher-layer protocol, typically the Logical Link Control (LLC) layer, in the form of a block of data known as the MAC service data unit (MSDU). In general, the MAC layer performs the following functions:

• On transmission, assemble data into a frame, known as a MAC protocol data unit (MPDU) with address and error-detection fields.

• On reception, disassemble frame, and perform address recognition and error detection.

• Govern access to the LAN transmission medium.

The exact format of the MPDU differs somewhat for the various MAC proto-cols in use. In general, all of the MPDUs have a format similar to that of Figure 18.4. The fields of this frame are as follows.

Logical LinkControl

Medium AccessControl

Physical

Encoding/decoding of signalsBit transmission/receptionTransmission medium

Assemble data into frameAddressingError detectionMedium access

Flow controlError control

General IEEE 802functions

Specific IEEE 802.11functions

Frequency band definitionWireless signal encoding

Reliable data deliveryWireless access control protocols

Figure 18.3 IEEE 802.11 Protocol Stack


• MAC Control: This field contains any protocol control information needed for the functioning of the MAC protocol. For example, a priority level could be indicated here.

• Destination MAC Address: The destination physical address on the LAN for this MPDU.

• Source MAC Address: The source physical address on the LAN for this MPDU.

• MAC Service Data Unit: The data from the next higher layer.

• CRC: The cyclic redundancy check field; also known as the Frame Check Sequence (FCS) field. This is an error-detecting code, such as that which is used in other data-link control protocols. The CRC is calculated based on the bits in the entire MPDU. The sender calculates the CRC and adds it to the frame. The receiver performs the same calculation on the incoming MPDU and compares that calculation to the CRC field in that incoming MPDU. If the two values don’t match, then one or more bits have been altered in transit.

The fields preceding the MSDU field are referred to as the MAC header, and the field following the MSDU field is referred to as the MAC trailer. The header and trailer contain control information that accompany the data field and that are used by the MAC protocol.

logical link conTrol In most data-link control protocols, the data-link protocol entity is responsible not only for detecting errors using the CRC, but for recovering from those errors by retransmitting damaged frames. In the LAN protocol archi-tecture, these two functions are split between the MAC and LLC layers. The MAC layer is responsible for detecting errors and discarding any frames that contain er-rors. The LLC layer optionally keeps track of which frames have been successfully received and retransmits unsuccessful frames.

IEEE 802.11 Network Components and Architectural Model

Figure 18.5 illustrates the model developed by the 802.11 working group. The small-est building block of a wireless LAN is a basic service set (BSS), which consists of wireless stations executing the same MAC protocol and competing for access to the same shared wireless medium. A BSS may be isolated, or it may connect to a back-bone distribution system (DS) through an access point (AP). The AP functions as a bridge and a relay point. In a BSS, client stations do not communicate directly with one another. Rather, if one station in the BSS wants to communicate with another station in the same BSS, the MAC frame is first sent from the originating station to

MACControl

reliart CAMredaeh CAM

DestinationMAC Address

SourceMAC Address MAC Service Data Unit (MSDU) CRC

Figure 18.4 General IEEE 802 MPDU Format


the AP and then from the AP to the destination station. Similarly, a MAC frame from a station in the BSS to a remote station is sent from the local station to the AP and then relayed by the AP over the DS on its way to the destination station. The BSS generally corresponds to what is referred to as a cell in the literature. The DS can be a switch, a wired network, or a wireless network.

When all the stations in the BSS are mobile stations that communicate directly with one another (not using an AP), the BSS is called an independent BSS (IBSS). An IBSS is typically an ad hoc network. In an IBSS, the stations all communicate directly, and no AP is involved.

A simple configuration is shown in Figure 18.5, in which each station belongs to a single BSS; that is, each station is within wireless range only of other stations within the same BSS. It is also possible for two BSSs to overlap geographically, so that a single station could participate in more than one BSS. Furthermore, the asso-ciation between a station and a BSS is dynamic. Stations may turn off, come within range, and go out of range.

An extended service set (ESS) consists of two or more basic service sets in-terconnected by a distribution system. The extended service set appears as a single logical LAN to the logical link control (LLC) level.

IEEE 802.11 Services

IEEE 802.11 defines nine services that need to be provided by the wireless LAN to achieve functionality equivalent to that which is inherent to wired LANs. Table 18.2 lists the services and indicates two ways of categorizing them.

STA 2

STA 3

STA4

STA 1

STA 6 STA 7

STA 8

AP 2

AP 1

Basic ServiceSet (BSS)

Basic ServiceSet (BSS)

Distribution System

Figure 18.5 IEEE 802.11 Extended Service Set


1. The service provider can be either the station or the DS. Station services are implemented in every 802.11 station, including AP stations. Distribution services are provided between BSSs; these services may be implemented in an AP or in another special-purpose device attached to the distribution system.

2. Three of the services are used to control IEEE 802.11 LAN access and confi-dentiality. Six of the services are used to support delivery of MSDUs between stations. If the MSDU is too large to be transmitted in a single MPDU, it may be fragmented and transmitted in a series of MPDUs.

Following the IEEE 802.11 document, we next discuss the services in an order designed to clarify the operation of an IEEE 802.11 ESS network. MSDU delivery, which is the basic service, already has been mentioned. Services related to security are introduced in Section 18.4.

diSTribuTion of meSSageS WiThin a dS The two services involved with the dis-tribution of messages within a DS are distribution and integration. Distribution is the primary service used by stations to exchange MPDUs when the MPDUs must traverse the DS to get from a station in one BSS to a station in another BSS. For example, suppose a frame is to be sent from station 2 (STA 2) to station 7 (STA 7) in Figure 18.5. The frame is sent from STA 2 to AP 1, which is the AP for this BSS. The AP gives the frame to the DS, which has the job of directing the frame to the AP associated with STA 7 in the target BSS. AP 2 receives the frame and forwards it to STA 7. How the message is transported through the DS is beyond the scope of the IEEE 802.11 standard.

If the two stations that are communicating are within the same BSS, then the distribution service logically goes through the single AP of that BSS.

The integration service enables transfer of data between a station on an IEEE 802.11 LAN and a station on an integrated IEEE 802.x LAN. The term integrated refers to a wired LAN that is physically connected to the DS and whose stations may be logically connected to an IEEE 802.11 LAN via the integration service. The integration service takes care of any address translation and media conversion logic required for the exchange of data.

Table 18.2 IEEE 802.11 Services

Service Provider Used to support

Association Distribution system MSDU delivery

Authentication Station LAN access and security

Deauthentication Station LAN access and security

Disassociation Distribution system MSDU delivery

Distribution Distribution system MSDU delivery

Integration Distribution system MSDU delivery

MSDU delivery Station MSDU delivery

Privacy Station LAN access and security

Reassociation Distribution system MSDU delivery


aSSociaTion-relaTed ServiceS The primary purpose of the MAC layer is to transfer MSDUs between MAC entities; this purpose is fulfilled by the distribution service. For that service to function, it requires information about stations within the ESS that is provided by the association-related services. Before the distribution service can deliver data to or accept data from a station, that station must be associated. Before looking at the concept of association, we need to describe the concept of mobility. The standard defines three transition types, based on mobility:

• No transition: A station of this type is either stationary or moves only within the direct communication range of the communicating stations of a single BSS.

• BSS transition: This is defined as a station movement from one BSS to an-other BSS within the same ESS. In this case, delivery of data to the station requires that the addressing capability be able to recognize the new location of the station.

• ESS transition: This is defined as a station movement from a BSS in one ESS to a BSS within another ESS. This case is supported only in the sense that the station can move. Maintenance of upper-layer connections supported by 802.11 cannot be guaranteed. In fact, disruption of service is likely to occur.

To deliver a message within a DS, the distribution service needs to know where the destination station is located. Specifically, the DS needs to know the identity of the AP to which the message should be delivered in order for that mes-sage to reach the destination station. To meet this requirement, a station must maintain an association with the AP within its current BSS. Three services relate to this requirement:

• Association: Establishes an initial association between a station and an AP. Before a station can transmit or receive frames on a wireless LAN, its identity and address must be known. For this purpose, a station must establish an as-sociation with an AP within a particular BSS. The AP can then communicate this information to other APs within the ESS to facilitate routing and delivery of addressed frames.

• Reassociation: Enables an established association to be transferred from one AP to another, allowing a mobile station to move from one BSS to another.

• Disassociation: A notification from either a station or an AP that an existing association is terminated. A station should give this notification before leaving an ESS or shutting down. However, the MAC management facility protects itself against stations that disappear without notification.

18.4 ieee 802.11i WireLess Lan security

There are two characteristics of a wired LAN that are not inherent in a wireless LAN.

1. In order to transmit over a wired LAN, a station must be physically connected to the LAN. On the other hand, with a wireless LAN, any station within radio

18.4 / ieee 802.11i Wireless laN security 573

range of the other devices on the LAN can transmit. In a sense, there is a form of authentication with a wired LAN in that it requires some positive and pre-sumably observable action to connect a station to a wired LAN.

2. Similarly, in order to receive a transmission from a station that is part of a wired LAN, the receiving station also must be attached to the wired LAN. On the other hand, with a wireless LAN, any station within radio range can receive. Thus, a wired LAN provides a degree of privacy, limiting reception of data to stations connected to the LAN.

These differences between wired and wireless LANs suggest the increased need for robust security services and mechanisms for wireless LANs. The original 802.11 specification included a set of security features for privacy and authenti-cation that were quite weak. For privacy, 802.11 defined the Wired Equivalent Privacy (WEP) algorithm. The privacy portion of the 802.11 standard contained major weaknesses. Subsequent to the development of WEP, the 802.11i task group has developed a set of capabilities to address the WLAN security issues. In order to accelerate the introduction of strong security into WLANs, the Wi-Fi Alliance promulgated Wi-Fi Protected Access (WPA) as a Wi-Fi standard. WPA is a set of security mechanisms that eliminates most 802.11 security issues and was based on the current state of the 802.11i standard. The final form of the 802.11i standard is referred to as Robust Security Network (RSN). The Wi-Fi Alliance certifies vendors in compliance with the full 802.11i specification under the WPA2 program.

The RSN specification is quite complex, and occupies 145 pages of the 2012 IEEE 802.11 standard. In this section, we provide an overview.

IEEE 802.11i Services

The 802.11i RSN security specification defines the following services.

• Authentication: A protocol is used to define an exchange between a user and an AS that provides mutual authentication and generates temporary keys to be used between the client and the AP over the wireless link.

• Access control:1 This function enforces the use of the authentication function, routes the messages properly, and facilitates key exchange. It can work with a variety of authentication protocols.

• Privacy with message integrity: MAC-level data (e.g., an LLC PDU) are en-crypted along with a message integrity code that ensures that the data have not been altered.

Figure 18.6a indicates the security protocols used to support these services, while Figure 18.6b lists the cryptographic algorithms used for these services.

1In this context, we are discussing access control as a security function. This is a different function than media access control (MAC) as described in Section 18.3. Unfortunately, the literature and the standards use the term access control in both contexts.


IEEE 802.11i Phases of Operation

The operation of an IEEE 802.11i RSN can be broken down into five distinct phases of operation. The exact nature of the phases will depend on the configuration and the end points of the communication. Possibilities include (see Figure 18.5):

1. Two wireless stations in the same BSS communicating via the access point (AP) for that BSS.

2. Two wireless stations (STAs) in the same ad hoc IBSS communicating directly with each other.

3. Two wireless stations in different BSSs communicating via their respective APs across a distribution system.

4. A wireless station communicating with an end station on a wired network via its AP and the distribution system.

Access Control

Serv

ices

Pro

toco

lsSe

rvic

esA

lgor

ithm

s

IEEE 802.1Port-based

Access Control

ExtensibleAuthenticationProtocol (EAP)

Authenticationand Key

Generation

(a) Services and protocols

Confidentiality, DataOrigin Authentication

and Integrity andReplay Protection

TKIP CCMP

Robust Security Network (RSN)

Confidentiality

TKIP(Michael

MIC)

CCM(AES-CBC-MAC)

CCM(AES-CTR)

NISTKey

Wrap

HMAC-MD5

HMAC-SHA-1

Integrity andData Origin

Authentication

(b) Cryptographic algorithms

KeyGeneration

TKIP(RC4)

Robust Security Network (RSN)

HMAC-SHA-1

RFC1750

CBC-MAC = Cipher Block Chaining Message Authentication Code (MAC)CCM = Counter Mode with Cipher Block Chaining Message Authentication Code CCMP = Counter Mode with Cipher Block Chaining MAC ProtocolTKIP = Temporal Key Integrity Protocol

Figure 18.6 Elements of IEEE 802.11i

IEEE 802.11i security is concerned only with secure communication between the STA and its AP. In case 1 in the preceding list, secure communication is assured if each STA establishes secure communications with the AP. Case 2 is similar, with the AP functionality residing in the STA. For case 3, security is not provided across the distribution system at the level of IEEE 802.11, but only within each BSS. End-to-end security (if required) must be provided at a higher layer. Similarly, in case 4, security is only provided between the STA and its AP.

With these considerations in mind, Figure 18.7 depicts the five phases of op-eration for an RSN and maps them to the network components involved. One new component is the authentication server (AS). The rectangles indicate the exchange of sequences of MPDUs. The five phases are defined as follows.

• Discovery: An AP uses messages called Beacons and Probe Responses to ad-vertise its IEEE 802.11i security policy. The STA uses these to identify an AP for a WLAN with which it wishes to communicate. The STA associates with the AP, which it uses to select the cipher suite and authentication mechanism when the Beacons and Probe Responses present a choice.

• Authentication: During this phase, the STA and AS prove their identities to each other. The AP blocks non-authentication traffic between the STA and AS until the authentication transaction is successful. The AP does not partici-pate in the authentication transaction other than forwarding traffic between the STA and AS.

Phase 1 - Discovery

STA AP AS End Station

Phase 5 - Connection Termination

Phase 3 - Key Management

Phase 4 - Protected Data Transfer

Phase 2 - Authentication

Figure 18.7 IEEE 802.11i Phases of Operation



• Key generation and distribution: The AP and the STA perform several opera-tions that cause cryptographic keys to be generated and placed on the AP and the STA. Frames are exchanged between the AP and STA only.

• Protected data transfer: Frames are exchanged between the STA and the end station through the AP. As denoted by the shading and the encryption module icon, secure data transfer occurs between the STA and the AP only; security is not provided end-to-end.

• Connection termination: The AP and STA exchange frames. During this phase, the secure connection is torn down and the connection is restored to the original state.

Discovery Phase

We now look in more detail at the RSN phases of operation, beginning with the discovery phase, which is illustrated in the upper portion of Figure 18.8. The pur-pose of this phase is for an STA and an AP to recognize each other, agree on a set of security capabilities, and establish an association for future communication using those security capabilities.

SecuriTy caPabiliTieS During this phase, the STA and AP decide on specific tech-niques in the following areas:

• Confidentiality and MPDU integrity protocols for protecting unicast traffic (traffic only between this STA and AP)

• Authentication method

• Cryptography key management approach

Confidentiality and integrity protocols for protecting multicast/broadcast traf-fic are dictated by the AP, since all STAs in a multicast group must use the same protocols and ciphers. The specification of a protocol, along with the chosen key length (if variable) is known as a cipher suite. The options for the confidentiality and integrity cipher suite are

• WEP, with either a 40-bit or 104-bit key, which allows backward compatibility with older IEEE 802.11 implementations

• TKIP

• CCMP

• Vendor-specific methods

The other negotiable suite is the authentication and key management (AKM) suite, which defines (1) the means by which the AP and STA perform mutual au-thentication and (2) the means for deriving a root key from which other keys may be generated. The possible AKM suites are

• IEEE 802.1X

• Pre-shared key (no explicit authentication takes place and mutual authentica-tion is implied if the STA and AP share a unique secret key)

• Vendor-specific methods

mPdu exchange The discovery phase consists of three exchanges.

• Network and security capability discovery: During this exchange, STAs dis-cover the existence of a network with which to communicate. The AP either periodically broadcasts its security capabilities (not shown in figure), indicated by RSN IE (Robust Security Network Information Element), in a specific channel through the Beacon frame; or responds to a station’s Probe Request through a Probe Response frame. A wireless station may discover available access points and corresponding security capabilities by either passively moni-toring the Beacon frames or actively probing every channel.

• Open system authentication: The purpose of this frame sequence, which provides no security, is simply to maintain backward compatibility with the

STA AP AS

Probe requestStation sends a requestto join network AP sends possible

security parameter(security capabilities setper the security policy)

AP performsnull authentication

AP sends the associatedsecurity parameters

Station sends arequest to perform

null authentication

Station sends a request toassociate with AP with

security parameters

Station sets selectedsecurity parameters

Open systemauthentication request

Probe response

802.1X EAP request

Access request(EAP request)

802.1X EAP response

Accept/EAP-successkey material

802.1X EAP success

Association request

Association response

Open systemauthentication response

802.1X-controlled port blocked

802.1X-controlled port blocked

Extensible Authentication Protocol Exchange

Figure 18.8 IEEE 802.11i Phases of Operation: Capability Discovery, Authentication, and Association



IEEE 802.11 state machine, as implemented in existing IEEE 802.11 hard-ware. In essence, the two devices (STA and AP) simply exchange identifiers.

• Association: The purpose of this stage is to agree on a set of security capabili-ties to be used. The STA then sends an Association Request frame to the AP. In this frame, the STA specifies one set of matching capabilities (one authentication and key management suite, one pairwise cipher suite, and one group-key cipher suite) from among those advertised by the AP. If there is no match in capabilities between the AP and the STA, the AP refuses the Association Request. The STA blocks it too, in case it has associated with a rogue AP or someone is inserting frames illicitly on its channel. As shown in Figure 18.8, the IEEE 802.1X controlled ports are blocked, and no user traffic goes beyond the AP. The concept of blocked ports is explained subsequently.

Authentication Phase

As was mentioned, the authentication phase enables mutual authentication be-tween an STA and an authentication server (AS) located in the DS. Authentication is designed to allow only authorized stations to use the network and to provide the STA with assurance that it is communicating with a legitimate network.

ieee 802.1x acceSS conTrol aPProach IEEE 802.11i makes use of another stan-dard that was designed to provide access control functions for LANs. The standard is IEEE 802.1X, Port-Based Network Access Control. The authentication proto-col that is used, the Extensible Authentication Protocol (EAP), is defined in the IEEE 802.1X standard. IEEE 802.1X uses the terms supplicant, authenticator, and authentication server (AS). In the context of an 802.11 WLAN, the first two terms correspond to the wireless station and the AP. The AS is typically a separate device on the wired side of the network (i.e., accessible over the DS) but could also reside directly on the authenticator.

Before a supplicant is authenticated by the AS using an authentication proto-col, the authenticator only passes control or authentication messages between the supplicant and the AS; the 802.1X control channel is unblocked, but the 802.11 data channel is blocked. Once a supplicant is authenticated and keys are provided, the authenticator can forward data from the supplicant, subject to predefined access control limitations for the supplicant to the network. Under these circumstances, the data channel is unblocked.

As indicated in Figure 16.5, 802.1X uses the concepts of controlled and uncon-trolled ports. Ports are logical entities defined within the authenticator and refer to physical network connections. For a WLAN, the authenticator (the AP) may have only two physical ports: one connecting to the DS and one for wireless communica-tion within its BSS. Each logical port is mapped to one of these two physical ports. An uncontrolled port allows the exchange of PDUs between the supplicant and the other AS, regardless of the authentication state of the supplicant. A controlled port allows the exchange of PDUs between a supplicant and other systems on the LAN only if the current state of the supplicant authorizes such an exchange. IEEE 802.1X is covered in more detail in Chapter 16.

The 802.1X framework, with an upper-layer authentication protocol, fits nicely with a BSS architecture that includes a number of wireless stations and an AP. However, for an IBSS, there is no AP. For an IBSS, 802.11i provides a more complex solution that, in essence, involves pairwise authentication between stations on the IBSS.

mPdu exchange The lower part of Figure 18.8 shows the MPDU exchange dic-tated by IEEE 802.11 for the authentication phase. We can think of authentication phase as consisting of the following three phases.

• Connect to AS: The STA sends a request to its AP (the one with which it has an association) for connection to the AS. The AP acknowledges this request and sends an access request to the AS.

• EAP exchange: This exchange authenticates the STA and AS to each other. A number of alternative exchanges are possible, as explained subsequently.

• Secure key delivery: Once authentication is established, the AS gener-ates a master session key (MSK), also known as the Authentication, Authorization, and Accounting (AAA) key and sends it to the STA. As explained subsequently, all the cryptographic keys needed by the STA for secure communication with its AP are generated from this MSK. IEEE 802.11i does not prescribe a method for secure delivery of the MSK but relies on EAP for this. Whatever method is used, it involves the trans-mission of an MPDU containing an encrypted MSK from the AS, via the AP, to the AS.

eaP exchange As mentioned, there are a number of possible EAP exchanges that can be used during the authentication phase. Typically, the message flow between STA and AP employs the EAP over LAN (EAPOL) protocol, and the message flow between the AP and AS uses the Remote Authentication Dial In User Service (RADIUS) protocol, although other options are available for both STA-to-AP and AP-to-AS exchanges. [FRAN07] provides the following summary of the authenti-cation exchange using EAPOL and RADIUS.

1. The EAP exchange begins with the AP issuing an EAP-Request/Identity frame to the STA.

2. The STA replies with an EAP-Response/Identity frame, which the AP receives over the uncontrolled port. The packet is then encapsulated in RADIUS over EAP and passed on to the RADIUS server as a RADIUS-Access-Request packet.

3. The AAA server replies with a RADIUS-Access-Challenge packet, which is passed on to the STA as an EAP-Request. This request is of the appropriate authentication type and contains relevant challenge information.

4. The STA formulates an EAP-Response message and sends it to the AS. The response is translated by the AP into a Radius-Access-Request with the re-sponse to the challenge as a data field. Steps 3 and 4 may be repeated multiple times, depending on the EAP method in use. For TLS tunneling methods, it is common for authentication to require 10 to 20 round trips.



5. The AAA server grants access with a Radius-Access-Accept packet. The AP issues an EAP-Success frame. (Some protocols require confirmation of the EAP success inside the TLS tunnel for authenticity validation.) The controlled port is authorized, and the user may begin to access the network.

Note from Figure 18.8 that the AP controlled port is still blocked to general user traffic. Although the authentication is successful, the ports remain blocked until the temporal keys are installed in the STA and AP, which occurs during the 4-Way Handshake.

Key Management Phase

During the key management phase, a variety of cryptographic keys are generated and distributed to STAs. There are two types of keys: pairwise keys used for com-munication between an STA and an AP and group keys used for multicast com-munication. Figure 18.9, based on [FRAN07], shows the two key hierarchies, and Table 18.3 defines the individual keys.

PairWiSe keyS Pairwise keys are used for communication between a pair of de-vices, typically between an STA and an AP. These keys form a hierarchy beginning with a master key from which other keys are derived dynamically and used for a limited period of time.

At the top level of the hierarchy are two possibilities. A pre-shared key (PSK) is a secret key shared by the AP and a STA and installed in some fashion outside the scope of IEEE 802.11i. The other alternative is the master session key (MSK), also known as the AAAK, which is generated using the IEEE 802.1X protocol dur-ing the authentication phase, as described previously. The actual method of key generation depends on the details of the authentication protocol used. In either case (PSK or MSK), there is a unique key shared by the AP with each STA with which it communicates. All the other keys derived from this master key are also unique between an AP and an STA. Thus, each STA, at any time, has one set of keys, as depicted in the hierarchy of Figure 18.9a, while the AP has one set of such keys for each of its STAs.

The pairwise master key (PMK) is derived from the master key. If a PSK is used, then the PSK is used as the PMK; if a MSK is used, then the PMK is derived from the MSK by truncation (if necessary). By the end of the authentication phase, marked by the 802.1X EAP Success message (Figure 18.8), both the AP and the STA have a copy of their shared PMK.

The PMK is used to generate the pairwise transient key (PTK), which in fact consists of three keys to be used for communication between an STA and AP after they have been mutually authenticated. To derive the PTK, the HMAC-SHA-1 function is applied to the PMK, the MAC addresses of the STA and AP, and nonces generated when needed. Using the STA and AP addresses in the generation of the PTK provides protection against session hijacking and impersonation; using nonces provides additional random keying material.

The three parts of the PTK are as follows.

• EAP Over LAN (EAPOL) Key Confirmation Key (EAPOL-KCK): Supports the integrity and data origin authenticity of STA-to-AP control frames during

operational setup of an RSN. It also performs an access control function: proof-of-possession of the PMK. An entity that possesses the PMK is autho-rized to use the link.

• EAPOL Key Encryption Key (EAPOL-KEK): Protects the confidentiality of keys and other data during some RSN association procedures.

• Temporal Key (TK): Provides the actual protection for user traffic.

grouP keyS Group keys are used for multicast communication in which one STA sends MPDU’s to multiple STAs. At the top level of the group key hierarchy is the group master key (GMK). The GMK is a key-generating key used with other inputs

Out-of-band path EAP method path

Pre-shared key

EAPOL key confirmation key EAPOL key encryption key Temporal key

PSK

256 bits

384 bits (CCMP)512 bits (TKIP)

128 bits (CCMP)256 bits (TKIP)

40 bits, 104 bits (WEP)128 bits (CCMP)256 bits (TKIP)

256 bits

128 bits

No modificationLegend

Possible truncationPRF (pseudo randomfunction) usingHMAC-SHA-1

128 bits

User-definedcryptoid

EAPauthentication

Following EAP authenticationor PSK

During 4-way handshake

These keys arecomponents of the PTK

≥ 256 bits

PMK

KCK

PTK

KTKEK

AAAK or MSK

Pairwise master key

(b) Group key hierarchy

(a) Pairwise key hierarchy

AAA key

Pairwise transient key

256 bits Changes periodicallyor if compromised

Changes based onpolicy (dissociation,deauthentication)

GMK (generated by AS)

GTK

Group master key

Group temporal key

Figure 18.9 IEEE 802.11i Key Hierarchies



to derive the group temporal key (GTK). Unlike the PTK, which is generated using material from both AP and STA, the GTK is generated by the AP and transmit-ted to its associated STAs. Exactly how this GTK is generated is undefined. IEEE 802.11i, however, requires that its value is computationally indistinguishable from random. The GTK is distributed securely using the pairwise keys that are already established. The GTK is changed every time a device leaves the network.

PairWiSe key diSTribuTion The upper part of Figure 18.10 shows the MPDU exchange for distributing pairwise keys. This exchange is known as the 4-way hand-shake. The STA and AP use this handshake to confirm the existence of the PMK, verify the selection of the cipher suite, and derive a fresh PTK for the following data session. The four parts of the exchange are as follows.

Table 18.3 IEEE 802.11i Keys for Data Confidentiality and Integrity Protocols

Abbrev-iation Name Description / Purpose Size (bits) Type

AAA Key Authentication, Accounting, and Authorization Key

Used to derive the PMK. Used with the IEEE 802.1X authen-tication and key management approach. Same as MMSK.

Ú 256 Key generation key, root key

PSK Pre-shared Key Becomes the PMK in pre-shared key environments.

256 Key generation key, root key

PMK Pairwise Master Key Used with other inputs to derive the PTK.

256 Key generation key

GMK Group Master Key Used with other inputs to derive the GTK.

128 Key generation key

PTK Pair-wise Transient Key

Derived from the PMK. Comprises the EAPOL-KCK, EAPOL-KEK, and TK and (for TKIP) the MIC key.

512 (TKIP)384 (CCMP)

Composite key

TK Temporal Key Used with TKIP or CCMP to pro-vide confidentiality and integrity protection for unicast user traffic.


Traffic key

GTK Group Temporal Key

Derived from the GMK. Used to provide confidentiality and integrity protection for multicast/broadcast user traffic.


40,104 (WEP)

Traffic key

MIC Key Message Integrity Code Key

Used by TKIP’s Michael MIC to provide integrity protection of messages.

64 Message integrity key

EAPOL-KCK

EAPOL-Key Confirmation Key

Used to provide integrity protec-tion for key material distributed during the 4-Way Handshake.

128 Message integrity key

EAPOL-KEK

EAPOL-Key Encryption Key

Used to ensure the confidentiality of the GTK and other key mate-rial in the 4-Way Handshake.

128 Traffic key / key encryption key

WEP Key Wired Equivalent Privacy Key

Used with WEP. 40,104 Traffic key

• AP S STA: Message includes the MAC address of the AP and a nonce (Anonce)

• STA S AP: The STA generates its own nonce (Snonce) and uses both nonces and both MAC addresses, plus the PMK, to generate a PTK. The STA then sends a message containing its MAC address and Snonce, enabling the AP to generate the same PTK. This message includes a message integrity code (MIC)2 using HMAC-MD5 or HMAC-SHA-1-128. The key used with the MIC is KCK.

STA AP

Message 1 delivers a nonce tothe STA so that it can generatethe PTK.

Message 1 delivers a new GTK tothe STA. The GTK is encryptedbefore it is sent and the entiremessage is integrity protected.

The AP installs the GTK.

Message 3 demonstrates tothe STA that the authenticatoris alive, ensures that the PTK isfresh (new) and that there is noman-in-the-middle.

Message 2 delivers another nonce to theAP so that it can also generate thePTK. It demonstrates to the AP thatthe STA is alive, ensures that thePTK is fresh (new) and that there is noman-in-the-middle.

The STA decrypts the GTKand installs it for use.

Message 2 is delivered to the AP. This frame serves only asan acknowledgment to the AP.

Message 4 serves as an acknowledgment toMessage 3. It serves no cryptographicfunction. This message also ensures thereliable start of the group key handshake.

Message 2EAPOL-key (Snonce,

Unicast, MIC)

Message 1EAPOL-key (Anonce, Unicast)

Message 1EAPOL-key (GTK, MIC)

Message 4EAPOL-key (Unicast, MIC)

Message 2EAPOL-key (MIC)

Message 3EAPOL-key (Install PTK,

Unicast, MIC)

AP’s 802.1X-controlled port blocked

AP’s 802.1X-controlled port unblocked for unicast traffic

Figure 18.10 IEEE 802.11i Phases of Operation: Four-Way Handshake and Group Key Handshake

2While MAC is commonly used in cryptography to refer to a Message Authentication Code, the term MIC is used instead in connection with 802.11i because MAC has another standard meaning, Media Access Control, in networking.



• AP S STA: The AP is now able to generate the PTK. The AP then sends a message to the STA, containing the same information as in the first message, but this time including a MIC.

• STA S AP: This is merely an acknowledgment message, again protected by a MIC.

grouP key diSTribuTion For group key distribution, the AP generates a GTK and distributes it to each STA in a multicast group. The two-message exchange with each STA consists of the following:

• AP S STA: This message includes the GTK, encrypted either with RC4 or with AES. The key used for encryption is KEK, using a key wrapping algo-rithm (as discussed in Chapter 12). A MIC value is appended.

• STA S AP: The STA acknowledges receipt of the GTK. This message in-cludes a MIC value.

Protected Data Transfer Phase

IEEE 802.11i defines two schemes for protecting data transmitted in 802.11 MPDUs: the Temporal Key Integrity Protocol (TKIP), and the Counter Mode-CBC MAC Protocol (CCMP).

TkiP TKIP is designed to require only software changes to devices that are imple-mented with the older wireless LAN security approach called Wired Equivalent Privacy (WEP). TKIP provides two services:

• Message integrity: TKIP adds a message integrity code (MIC) to the 802.11 MAC frame after the data field. The MIC is generated by an algorithm, called Michael, that computes a 64-bit value using as input the source and destina-tion MAC address values and the Data field, plus key material.

• Data confidentiality: Data confidentiality is provided by encrypting the MPDU plus MIC value using RC4.

The 256-bit TK (Figure 18.9) is employed as follows. Two 64-bit keys are used with the Michael message digest algorithm to produce a message integrity code. One key is used to protect STA-to-AP messages, and the other key is used to pro-tect AP-to-STA messages. The remaining 128 bits are truncated to generate the RC4 key used to encrypt the transmitted data.

For additional protection, a monotonically increasing TKIP sequence counter (TSC) is assigned to each frame. The TSC serves two purposes. First, the TSC is included with each MPDU and is protected by the MIC to protect against replay attacks. Second, the TSC is combined with the session TK to produce a dynamic en-cryption key that changes with each transmitted MPDU, thus making cryptanalysis more difficult.

ccmP CCMP is intended for newer IEEE 802.11 devices that are equipped with the hardware to support this scheme. As with TKIP, CCMP provides two services:

• Message integrity: CCMP uses the cipher block chaining message authentica-tion code (CBC-MAC), described in Chapter 12.

• Data confidentiality: CCMP uses the CTR block cipher mode of operation with AES for encryption. CTR is described in Chapter 6.

The same 128-bit AES key is used for both integrity and confidentiality. The scheme uses a 48-bit packet number to construct a nonce to prevent replay attacks.

The IEEE 802.11i Pseudorandom Function

At a number of places in the IEEE 802.11i scheme, a pseudorandom function (PRF) is used. For example, it is used to generate nonces, to expand pairwise keys, and to generate the GTK. Best security practice dictates that different pseudorandom number streams be used for these different purposes. However, for implementa-tion efficiency, we would like to rely on a single pseudorandom number generator function.

The PRF is built on the use of HMAC-SHA-1 to generate a pseudorandom bit stream. Recall that HMAC-SHA-1 takes a message (block of data) and a key of length at least 160 bits and produces a 160-bit hash value. SHA-1 has the property that the change of a single bit of the input produces a new hash value with no appar-ent connection to the preceding hash value. This property is the basis for pseudo-random number generation.

The IEEE 802.11i PRF takes four parameters as input and produces the desired number of random bits. The function is of the form PRF(K, A, B, Len), where

K = a secret keyA = a text string specific to the application (e.g., nonce generation or pair-

wise key expansion)B = some data specific to each caseLen = desired number of pseudorandom bits

For example, for the pairwise transient key for CCMP:

PTK = PRF(PMK,"Pairwise key expansion",min(AP- Addr,STA-Addr) } max(AP-Addr,STA-Addr) } min (Anonce,Snonce) } max(Anonce,Snonce),384)

So, in this case, the parameters are

K = PMKA = the text string “Pairwise key expansion”B = a sequence of bytes formed by concatenating the two MAC addresses

and the two noncesLen = 384 bits

Similarly, a nonce is generated by

Nonce = PRF(Random Number,"InitCounter",MAC } Time, 256)



where Time is a measure of the network time known to the nonce generator. The group temporal key is generated by

GTK = PRF(GMK,"Group key expansion", MAC } Gnonce, 256)

Figure 18.11 illustrates the function PRF(K, A, B, Len). The parameter K serves as the key input to HMAC. The message input consists of four items concat-enated together: the parameter A, a byte with value 0, the parameter B, and a coun-ter i. The counter is initialized to 0. The HMAC algorithm is run once, producing a 160-bit hash value. If more bits are required, HMAC is run again with the same inputs, except that i is incremented each time until the necessary number of bits is generated. We can express the logic as

PRF(K, A, B, Len)

R d null string

for i d 0 to ((Len + 159)/160 − 1) do

R d R } HMAC-SHA-1(K, A } 0 } B } i)Return Truncate-to-Len(R, Len)

18.5 recOMMenDeD reaDing

[SOUP12] provides a good overview of mobile device threats and countermeasures. [BECH11] is a useful analysis of smartphone security issues.

The IEEE 802.11 and Wi-Fi specifications are covered in more detail in [STAL11]. [FRAN07] is an excellent, detailed treatment of IEEE 802.11i. [CHEN05a] provides an over-view of IEEE 802.11i.

HMAC-SHA-1

| |

K

A 0 B i

R = HMAC-SHA-1(K, A || 0 || B || i)

+ 1

Figure 18.11 IEEE 802.11i Pseudorandom Function

18.6 / key terMs, revieW QuestioNs, aND probleMs 587

18.6 Key terMs, revieW QuestiOns, anD PrObLeMs

Key Terms

4-way handshakeaccess point (AP)Alert Protocolbasic service set (BSS)Counter Mode-CBC MAC

Protocol (CCMP)distribution system (DS)extended service set (ESS)group keysHandshake ProtocolIEEE 802.1XIEEE 802.11

IEEE 802.11iindependent BSS (IBSS)logical link control

(LLC)media access control (MAC)MAC protocol data unit

(MPDU)MAC service data unit

(MSDU)message integrity code

(MIC)Michael

pairwise keyspseudorandom functionRobust Security Network

(RSN)Temporal Key Integrity

Protocol (TKIP)Wired Equivalent Privacy

(WEP)Wireless LAN (WLAN)Wi-FiWi-Fi Protected Access

(WPA)

BECH11 Becher, M., et al. “Mobile Security Catching Up? Revealing the Nuts and Bolts of the Security of Mobile Devices.” IEEE Symposium on Security and Privacy, May 2011.

CHEN05a Chen, J.; Jiang, M.; and Liu, Y. “Wireless LAN Security and IEEE 802.11i.” IEEE Wireless Communications, February 2005.

FRAN07 Frankel, S.; Eydt, B.; Owens, L.; and Scarfone, K. Establishing Wireless Robust Security Networks: A Guide to IEEE 802.11i. NIST Special Publication SP 800-97, February 2007.

SOUP12 Souppaya, M.; and Scarfone, K. Guidelines for Managing and Securing Mobile Devices in the Enterprise. NIST Special Publication SP 800-124, July 2012.

STAL11 Stallings, W. Data and Computer Communications, Ninth Edition. Upper Saddle River, NJ: Prentice Hall, 2011.

Review Questions 18.1 What is the basic building block of an 802.11 WLAN? 18.2 Define an extended service set. 18.3 List and briefly define IEEE 802.11 services. 18.4 Is a distribution system a wireless network? 18.5 How is the concept of an association related to that of mobility? 18.6 What security areas are addressed by IEEE 802.11i? 18.7 Briefly describe the four IEEE 802.11i phases of operation. 18.8 What is the difference between TKIP and CCMP?


Problems 18.1 In IEEE 802.11, open system authentication simply consists of two communications.

An authentication is requested by the client, which contains the station ID (typically the MAC address). This is followed by an authentication response from the AP/router containing a success or failure message. An example of when a failure may occur is if the client’s MAC address is explicitly excluded in the AP/router configuration.

a. What are the benefits of this authentication scheme? b. What are the security vulnerabilities of this authentication scheme? 18.2 Prior to the introduction of IEEE 802.11i, the security scheme for IEEE 802.11 was

Wired Equivalent Privacy (WEP). WEP assumed all devices in the network share a secret key. The purpose of the authentication scenario is for the STA to prove that it possesses the secret key. Authentication proceeds as shown in Figure 18.12. The STA sends a message to the AP requesting authentication. The AP issues a challenge, which is a sequence of 128 random bytes sent as plaintext. The STA encrypts the challenge with the shared key and returns it to the AP. The AP decrypts the incom-ing value and compares it to the challenge that it sent. If there is a match, the AP confirms that authentication has succeeded.

a. What are the benefits of this authentication scheme? b. This authentication scheme is incomplete. What is missing and why is this impor-

tant? Hint: The addition of one or two messages would fix the problem. c. What is a cryptographic weakness of this scheme? 18.3 For WEP, data integrity and data confidentiality are achieved using the RC4 stream

encryption algorithm. The transmitter of an MPDU performs the following steps, referred to as encapsulation:

1. The transmitter selects an initial vector (IV) value. 2. The IV value is concatenated with the WEP key shared by transmitter and

receiver to form the seed, or key input, to RC4. 3. A 32-bit cyclic redundancy check (CRC) is computed over all the bits of the MAC

data field and appended to the data field. The CRC is a common error-detection code used in data link control protocols. In this case, the CRC serves as a integrity check value (ICV).

STA AP

RequestStation sends a requestfor authentication

AP sends challenge messagecontaining 128-bit randomnumber

AP decrypts challenge response.If match, send authenticationsuccess message

Station respondswith encrypted version

of challenge number

Response

Challenge

Success

Figure 18.12 WEP Authentication

18.6 / key terMs, revieW QuestioNs, aND probleMs 589

4. The result of step 3 is encrypted using RC4 to form the ciphertext block. 5. The plaintext IV is prepended to the ciphertext block to form the encapsulated

MPDU for transmission. a. Draw a block diagram that illustrates the encapsulation process. b. Describe the steps at the receiver end to recover the plaintext and perform the

integrity check. c. Draw a block diagram that illustrates part b. 18.4 A potential weakness of the CRC as an integrity check is that it is a linear function.

This means that you can predict which bits of the CRC are changed if a single bit of the message is changed. Furthermore, it is possible to determine which combination of bits could be flipped in the message so that the net result is no change in the CRC. Thus, there are a number of combinations of bit flippings of the plaintext message that leave the CRC unchanged, so message integrity is defeated. However, in WEP, if an attacker does not know the encryption key, the attacker does not have access to the plaintext, only to the ciphertext block. Does this mean that the ICV is protected from the bit flipping attack? Explain.

ElEctronic Mail SEcurity19.1 Pretty Good Privacy

Notation Operational Description

19.2 S/MIME

RFC 5322 Multipurpose Internet Mail Extensions S/MIME Functionality S/MIME Messages S/MIME Certificate Processing Enhanced Security Services

19.3 DomainKeys Identified Mail

Internet Mail Architecture E-mail Threats DKIM Strategy DKIM Functional Flow



Appendix 19A Radix-64 Conversion

Chapter

590590

19.1 / Pretty Good Privacy 591

Learning Objectives


u Present an overview of the operation of PGP (Pretty Good Privacy).

u Present an overview of MIME (Multipurpose Internet Mail Extension).

u Understand the functionality of S/MIME (Secure/Multipurpose Internet Mail Extension) and the security threats it addresses.

u Summarize the key functional components of the Internet mail architecture.

u Understand the role of DKIM (DomainKeys Identified Mail).

In virtually all distributed environments, electronic mail is the most heavily used network-based application. Users expect to be able to, and do, send e-mail to oth-ers who are connected directly or indirectly to the Internet, regardless of host operat-ing system or communications suite. With the explosively growing reliance on e-mail, there grows a demand for authentication and confidentiality services. Two schemes stand out as approaches that enjoy widespread use: Pretty Good Privacy (PGP) and S/MIME. Both are examined in this chapter. The chapter closes with a discussion of DomainKeys Identified Mail.

19.1 Pretty gOOd Privacy

PGP is a remarkable phenomenon. Largely the effort of a single person, Phil Zimmermann, PGP provides a confidentiality and authentication service that can be used for electronic mail and file storage applications. In essence, Zimmermann has done the following:

1. Selected the best available cryptographic algorithms as building blocks.

2. Integrated these algorithms into a general-purpose application that is inde-pendent of operating system and processor and that is based on a small set of easy-to-use commands.

Despite the refusal of VADM Poindexter and LtCol North to appear, the Board’s access to other sources of information filled much of this gap. The FBI provided documents taken from the files of the National Security Advisor and relevant NSC staff members, including messages from the PROF system between VADM Poin-dexter and LtCol North. The PROF messages were conversations by computer, written at the time events occurred and presumed by the writers to be protected from disclosure. In this sense, they provide a first-hand, contemporaneous account of events.

—The Tower Commission Report to President Reagan on the Iran-Contra Affair, 1987

592 chaPter 19 / electronic Mail Security

3. Made the package and its documentation, including the source code, freely available via the Internet, bulletin boards, and commercial networks such as AOL (America On Line).

4. Entered into an agreement with a company (Viacrypt, now Network Associates) to provide a fully compatible, low-cost commercial version of PGP.

PGP has grown explosively and is now widely used. A number of reasons can be cited for this growth.

1. It is available free worldwide in versions that run on a variety of platforms, in-cluding Windows, UNIX, Macintosh, and many more. In addition, the commer-cial version satisfies users who want a product that comes with vendor support.

2. It is based on algorithms that have survived extensive public review and are considered extremely secure. Specifically, the package includes RSA, DSS, and Diffie-Hellman for public-key encryption; CAST-128, IDEA, and 3DES for symmetric encryption; and SHA-1 for hash coding.

3. It has a wide range of applicability, from corporations that wish to select and enforce a standardized scheme for encrypting files and messages to individuals who wish to communicate securely with others worldwide over the Internet and other networks.

4. It was not developed by, nor is it controlled by, any governmental or standards organization. For those with an instinctive distrust of “the establishment,” this makes PGP attractive.

5. PGP is now on an Internet standards track (RFC 3156; MIME Security with OpenPGP). Nevertheless, PGP still has an aura of an antiestablishment endeavor.

We begin with an overall look at the operation of PGP. Appendix P examines how cryptographic keys are created and stored, and address the vital issue of public-key management.

Notation

Most of the notation used in this chapter has been used before, but a few terms are new. It is perhaps best to summarize those at the beginning. The following symbols are used.

Ks = session key used in symmetric encryption scheme

PRa = private key of user A, used in public-key encryption scheme

PUa = public key of user A, used in public-key encryption scheme

EP = public-key encryption

DP = public-key decryption

EC = symmetric encryption

DC = symmetric decryption

H = hash function

} = concatenation


Z = compression using ZIP algorithmR64 = conversion to radix 64 ASCII format1

The PGP documentation often uses the term secret key to refer to a key paired with a public key in a public-key encryption scheme. As was mentioned earlier, this practice risks confusion with a secret key used for symmetric encryption. Hence, we use the term private key instead.

Operational Description

The actual operation of PGP, as opposed to the management of keys, consists of four services: authentication, confidentiality, compression, and e-mail compatibility (Table 19.1). We examine each of these in turn.

AuthenticAtion Figure 19.1a illustrates the digital signature service provided by PGP. This is the digital signature scheme discussed in Chapter 13 and illustrated in Figure 13.2. The sequence is as follows.

1. The sender creates a message.

2. SHA-1 is used to generate a 160-bit hash code of the message.

3. The hash code is encrypted with RSA using the sender’s private key, and the result is prepended to the message.

4. The receiver uses RSA with the sender’s public key to decrypt and recover the hash code.

5. The receiver generates a new hash code for the message and compares it with the decrypted hash code. If the two match, the message is accepted as authentic.

1The American Standard Code for Information Interchange (ASCII) is described in Appendix Q.

Table 19.1 Summary of PGP Services

Function Algorithms Used Description

Digital signature

DSS/SHA or RSA/SHA

A hash code of a message is created using SHA-1. This message digest is encrypted using DSS or RSA with the sender’s private key and included with the message.

Message encryption

CAST or IDEA or Three-key Triple DES with Diffie-Hellman

or RSA

A message is encrypted using CAST-128 or IDEA or 3DES with a one-time session key generated by the sender. The session key is encrypted using Diffie-Hellman or RSA with the recipient’s public key and included with the message.

Compression ZIPA message may be compressed for storage or transmission using ZIP.

E-mail compatibility

Radix-64 conversionTo provide transparency for e-mail applica-tions, an encrypted message may be converted to an ASCII string using radix-64 conversion.

594

M

(c) Confidentiality and authentication

H

M

H

DP

Compare

PUa

| |

PRa

EP Z

EP

PUb

| |EC

Ks

DC

DP

PRb

Z�1

M

(b) Confidentiality only

DP

PRb

DCM

EP

PUb

EC | |

Ks

Z Z�1

H||

PRa

EPM

(a) Authentication only

Z Z�1

H

DP

Compare

PUa

M

E[PRa, H(M)]

E[PRa, H(M)]


E[PUb, Ks]

E[PUb, Ks]

Figure 19.1 PGP Cryptographic Functions


The combination of SHA-1 and RSA provides an effective digital signature scheme. Because of the strength of RSA, the recipient is assured that only the pos-sessor of the matching private key can generate the signature. Because of the strength of SHA-1, the recipient is assured that no one else could generate a new message that matches the hash code and, hence, the signature of the original message.

As an alternative, signatures can be generated using DSS/SHA-1.Although signatures normally are found attached to the message or file that

they sign, this is not always the case: Detached signatures are supported. A detached signature may be stored and transmitted separately from the message it signs. This is useful in several contexts. A user may wish to maintain a separate signature log of all messages sent or received. A detached signature of an executable program can detect subsequent virus infection. Finally, detached signatures can be used when more than one party must sign a document, such as a legal contract. Each person’s signature is independent and therefore is applied only to the document. Otherwise, signatures would have to be nested, with the second signer signing both the docu-ment and the first signature, and so on.

confidentiAlity Another basic service provided by PGP is confidentiality, which is provided by encrypting messages to be transmitted or to be stored locally as files. In both cases, the symmetric encryption algorithm CAST-128 may be used. Alternatively, IDEA or 3DES may be used. The 64-bit cipher feedback (CFB) mode is used.

As always, one must address the problem of key distribution. In PGP, each symmetric key is used only once. That is, a new key is generated as a random 128-bit number for each message. Thus, although this is referred to in the documentation as a session key, it is in reality a one-time key. Because it is to be used only once, the session key is bound to the message and transmitted with it. To protect the key, it is encrypted with the receiver’s public key. Figure 19.1b illustrates the sequence, which can be described as follows.

1. The sender generates a message and a random 128-bit number to be used as a session key for this message only.

2. The message is encrypted using CAST-128 (or IDEA or 3DES) with the ses-sion key.

3. The session key is encrypted with RSA using the recipient’s public key and is prepended to the message.

4. The receiver uses RSA with its private key to decrypt and recover the session key.

5. The session key is used to decrypt the message.

As an alternative to the use of RSA for key encryption, PGP provides an op-tion referred to as Diffie-Hellman. As was explained in Chapter 10, Diffie-Hellman is a key exchange algorithm. In fact, PGP uses a variant of Diffie-Hellman that does provide encryption/decryption, known as ElGamal (Chapter 10).

Several observations may be made. First, to reduce encryption time, the com-bination of symmetric and public-key encryption is used in preference to simply using RSA or ElGamal to encrypt the message directly: CAST-128 and the other


symmetric algorithms are substantially faster than RSA or ElGamal. Second, the use of the public-key algorithm solves the session-key distribution problem, because only the recipient is able to recover the session key that is bound to the message. Note that we do not need a session-key exchange protocol of the type discussed in Chapter 14, because we are not beginning an ongoing session. Rather, each message is a one-time independent event with its own key. Furthermore, given the store-and-forward nature of electronic mail, the use of handshaking to assure that both sides have the same session key is not practical. Finally, the use of one-time symmetric keys strengthens what is already a strong symmetric encryption approach. Only a small amount of plaintext is encrypted with each key, and there is no relationship among the keys. Thus, to the extent that the public-key algorithm is secure, the entire scheme is secure. To this end, PGP provides the user with a range of key size options from 768 to 3072 bits (the DSS key for signatures is limited to 1024 bits).

confidentiAlity And AuthenticAtion As Figure 19.1c illustrates, both services may be used for the same message. First, a signature is generated for the plaintext message and prepended to the message. Then the plaintext message plus signature is encrypted using CAST-128 (or IDEA or 3DES), and the session key is encrypted using RSA (or ElGamal). This sequence is preferable to the opposite: encrypting the message and then generating a signature for the encrypted message. It is gen-erally more convenient to store a signature with a plaintext version of a message. Furthermore, for purposes of third-party verification, if the signature is performed first, a third party need not be concerned with the symmetric key when verifying the signature.

In summary, when both services are used, the sender first signs the message with its own private key, then encrypts the message with a session key, and finally encrypts the session key with the recipient’s public key.

compression As a default, PGP compresses the message after applying the sig-nature but before encryption. This has the benefit of saving space both for e-mail transmission and for file storage.

The placement of the compression algorithm, indicated by Z for compression and Z-1 for decompression in Figure 19.1, is critical.

1. The signature is generated before compression for two reasons:

a. It is preferable to sign an uncompressed message so that one can store only the uncompressed message together with the signature for future verifica-tion. If one signed a compressed document, then it would be necessary ei-ther to store a compressed version of the message for later verification or to recompress the message when verification is required.

b. Even if one were willing to generate dynamically a recompressed message for verification, PGP’s compression algorithm presents a difficulty. The algorithm is not deterministic; various implementations of the algorithm achieve different tradeoffs in running speed versus compression ratio and, as a result, produce different compressed forms. However, these different compression algorithms are interoperable because any version of the algo-rithm can correctly decompress the output of any other version. Applying


the hash function and signature after compression would constrain all PGP implementations to the same version of the compression algorithm.

2. Message encryption is applied after compression to strengthen cryptographic security. Because the compressed message has less redundancy than the origi-nal plaintext, cryptanalysis is more difficult.

The compression algorithm used is ZIP, which is described in Appendix O.

e-mAil compAtibility When PGP is used, at least part of the block to be transmit-ted is encrypted. If only the signature service is used, then the message digest is encrypted (with the sender’s private key). If the confidentiality service is used, the message plus signature (if present) are encrypted (with a one-time symmetric key). Thus, part or all of the resulting block consists of a stream of arbitrary 8-bit octets. However, many electronic mail systems only permit the use of blocks consisting of ASCII text. To accommodate this restriction, PGP provides the service of convert-ing the raw 8-bit binary stream to a stream of printable ASCII characters.

The scheme used for this purpose is radix-64 conversion. Each group of three octets of binary data is mapped into four ASCII characters. This format also ap-pends a CRC to detect transmission errors. See Appendix 19A for a description.

The use of radix 64 expands a message by 33%. Fortunately, the session key and signature portions of the message are relatively compact, and the plaintext mes-sage has been compressed. In fact, the compression should be more than enough to compensate for the radix-64 expansion. For example, [HELD96] reports an average compression ratio of about 2.0 using ZIP. If we ignore the relatively small signature and key components, the typical overall effect of compression and expansion of a file of length X would be 1.33 * 0.5 * X = 0.665 * X. Thus, there is still an over-all compression of about one-third.

One noteworthy aspect of the radix-64 algorithm is that it blindly converts the input stream to radix-64 format regardless of content, even if the input happens to be ASCII text. Thus, if a message is signed but not encrypted and the conversion is applied to the entire block, the output will be unreadable to the casual observer, which provides a certain level of confidentiality. As an option, PGP can be config-ured to convert to radix-64 format only the signature portion of signed plaintext messages. This enables the human recipient to read the message without using PGP. PGP would still have to be used to verify the signature.

Figure 19.2 shows the relationship among the four services so far discussed. On transmission (if it is required), a signature is generated using a hash code of the uncompressed plaintext. Then the plaintext (plus signature if present) is com-pressed. Next, if confidentiality is required, the block (compressed plaintext or compressed signature plus plaintext) is encrypted and prepended with the public-key-encrypted symmetric encryption key. Finally, the entire block is converted to radix-64 format.

On reception, the incoming block is first converted back from radix-64 format to binary. Then, if the message is encrypted, the recipient recovers the session key and decrypts the message. The resulting block is then decompressed. If the message is signed, the recipient recovers the transmitted hash code and compares it to its own calculation of the hash code.

598

(a) Generic transmission diagram (from A) (b) Generic reception diagram (to B)

Confidentialityrequired?

Confidentialityrequired?

Yes

No

Yes

No

X file

Convert to radix 64X R64[X]

Encrypt key, XX E(PUb, Ks) || E(Ks, X)

Generate signatureX signature || X

CompressX Z(X)

Decrypt key, XYes

No

Signaturerequired?

Signaturerequired?

Decompress

X Z�1(X)

Strip signature from Xverify signature

Yes

No

Ks D(PRb, E(PUb, Ks))X D(Ks, E(Ks, X))

Convert from radix 64X R64�1[X]

Figure 19.2 Transmission and Reception of PGP Messages

19.2 / S/MiMe 599

19.2 s/MiMe

Secure/Multipurpose Internet Mail Extension (S/MIME) is a security enhancement to the MIME Internet e-mail format standard based on technology from RSA Data Security. Although both PGP and S/MIME are on an IETF standards track, it ap-pears likely that S/MIME will emerge as the industry standard for commercial and organizational use, while PGP will remain the choice for personal e-mail security for many users. S/MIME is defined in a number of documents—most importantly RFCs 3370, 3850, 3851, and 3852.

To understand S/MIME, we need first to have a general understanding of the un-derlying e-mail format that it uses, namely MIME. But to understand the significance of MIME, we need to go back to the traditional e-mail format standard, RFC 822, which is still in common use. The most recent version of this format specification is RFC 5322 (Internet Message Format). Accordingly, this section first provides an intro-duction to these two earlier standards and then moves on to a discussion of S/MIME.

RFC 5322

RFC 5322 defines a format for text messages that are sent using electronic mail. It has been the standard for Internet-based text mail messages and remains in com-mon use. In the RFC 5322 context, messages are viewed as having an envelope and contents. The envelope contains whatever information is needed to accomplish transmission and delivery. The contents compose the object to be delivered to the recipient. The RFC 5322 standard applies only to the contents. However, the con-tent standard includes a set of header fields that may be used by the mail system to create the envelope, and the standard is intended to facilitate the acquisition of such information by programs.

The overall structure of a message that conforms to RFC 5322 is very simple. A message consists of some number of header lines (the header) followed by unre-stricted text (the body). The header is separated from the body by a blank line. Put differently, a message is ASCII text, and all lines up to the first blank line are as-sumed to be header lines used by the user agent part of the mail system.

A header line usually consists of a keyword, followed by a colon, followed by the keyword’s arguments; the format allows a long line to be broken up into several lines. The most frequently used keywords are From, To, Subject, and Date. Here is an example message:

Date: October 8, 2009 2:15:49 PM EDT

From: “William Stallings” <[email protected]>

Subject: The Syntax in RFC 5322

To: [email protected]

Cc: [email protected]

Hello. This section begins the actual

message body, which is delimited from the

message heading by a blank line.


Another field that is commonly found in RFC 5322 headers is Message-ID. This field contains a unique identifier associated with this message.

Multipurpose Internet Mail Extensions

Multipurpose Internet Mail Extension (MIME) is an extension to the RFC 5322 framework that is intended to address some of the problems and limitations of the use of Simple Mail Transfer Protocol (SMTP), defined in RFC 821, or some other mail transfer protocol and RFC 5322 for electronic mail. [PARZ06] lists the follow-ing limitations of the SMTP/5322 scheme.

1. SMTP cannot transmit executable files or other binary objects. A number of schemes are in use for converting binary files into a text form that can be used by SMTP mail systems, including the popular UNIX UUencode/UUdecode scheme. However, none of these is a standard or even a de facto standard.

2. SMTP cannot transmit text data that includes national language characters, because these are represented by 8-bit codes with values of 128 decimal or higher, and SMTP is limited to 7-bit ASCII.

3. SMTP servers may reject mail message over a certain size.

4. SMTP gateways that translate between ASCII and the character code EBCDIC do not use a consistent set of mappings, resulting in translation problems.

5. SMTP gateways to X.400 electronic mail networks cannot handle nontextual data included in X.400 messages.

6. Some SMTP implementations do not adhere completely to the SMTP stan-dards defined in RFC 821. Common problems include:

• Deletion, addition, or reordering of carriage return and linefeed

• Truncating or wrapping lines longer than 76 characters

• Removal of trailing white space (tab and space characters)

• Padding of lines in a message to the same length

• Conversion of tab characters into multiple space characters

MIME is intended to resolve these problems in a manner that is compatible with existing RFC 5322 implementations. The specification is provided in RFCs 2045 through 2049.

overview The MIME specification includes the following elements.

1. Five new message header fields are defined, which may be included in an RFC 5322 header. These fields provide information about the body of the message.

2. A number of content formats are defined, thus standardizing representations that support multimedia electronic mail.

3. Transfer encodings are defined that enable the conversion of any content for-mat into a form that is protected from alteration by the mail system.

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In this subsection, we introduce the five message header fields. The next two subsections deal with content formats and transfer encodings.

The five header fields defined in MIME are

• MIME-Version: Must have the parameter value 1.0. This field indicates that the message conforms to RFCs 2045 and 2046.

• Content-Type: Describes the data contained in the body with sufficient detail that the receiving user agent can pick an appropriate agent or mechanism to represent the data to the user or otherwise deal with the data in an appropriate manner.

• Content-Transfer-Encoding: Indicates the type of transformation that has been used to represent the body of the message in a way that is acceptable for mail transport.

• Content-ID: Used to identify MIME entities uniquely in multiple contexts.

• Content-Description: A text description of the object with the body; this is useful when the object is not readable (e.g., audio data).

Any or all of these fields may appear in a normal RFC 5322 header. A com-pliant implementation must support the MIME-Version, Content-Type, and Content-Transfer-Encoding fields; the Content-ID and Content-Description fields are optional and may be ignored by the recipient implementation.

mime content types The bulk of the MIME specification is concerned with the definition of a variety of content types. This reflects the need to provide standard-ized ways of dealing with a wide variety of information representations in a multi-media environment.

Table 19.2 lists the content types specified in RFC 2046. There are seven dif-ferent major types of content and a total of 15 subtypes. In general, a content type declares the general type of data, and the subtype specifies a particular format for that type of data.

For the text type of body, no special software is required to get the full meaning of the text aside from support of the indicated character set. The primary subtype is plain text, which is simply a string of ASCII characters or ISO 8859 characters. The enriched subtype allows greater formatting flexibility.

The multipart type indicates that the body contains multiple, independent parts. The Content-Type header field includes a parameter (called a boundary) that defines the delimiter between body parts. This boundary should not appear in any parts of the message. Each boundary starts on a new line and consists of two hy-phens followed by the boundary value. The final boundary, which indicates the end of the last part, also has a suffix of two hyphens. Within each part, there may be an optional ordinary MIME header.

Here is a simple example of a multipart message containing two parts—both consisting of simple text (taken from RFC 2046).

From: Nathaniel Borenstein <[email protected]>To: Ned Freed <[email protected]>Subject: Sample message


MIME-Version: 1.0Content-type: multipart/mixed; boundary="simple boundary"

This is the preamble. It is to be ignored, though it is a handy place for mail composers to include an explanatory note to non-MIME conformant readers.—simple boundary

This is implicitly typed plain ASCII text. It does NOT end with a linebreak.—simple boundaryContent-type: text/plain; charset=us-asciiThis is explicitly typed plain ASCII text. It DOES end with a linebreak.—simple boundary—This is the epilogue. It is also to be ignored.

There are four subtypes of the multipart type, all of which have the same over-all syntax. The multipart/mixed subtype is used when there are multiple indepen-dent body parts that need to be bundled in a particular order. For the multipart/parallel subtype, the order of the parts is not significant. If the recipient’s system is appropriate, the multiple parts can be presented in parallel. For example, a picture

Table 19.2 MIME Content Types

Type Subtype Description

Text Plain Unformatted text; may be ASCII or ISO 8859.

Enriched Provides greater format flexibility.

Multipart Mixed The different parts are independent but are to be transmitted together. They should be presented to the receiver in the order that they appear in the mail message.

Parallel Differs from Mixed only in that no order is defined for delivering the parts to the receiver.

Alternative The different parts are alternative versions of the same information. They are ordered in increasing faithfulness to the original, and the recipient’s mail system should display the “best” version to the user.

Digest Similar to Mixed, but the default type/subtype of each part is message/rfc822.

Message rfc822 The body is itself an encapsulated message that conforms to RFC 822.

Partial Used to allow fragmentation of large mail items, in a way that is transparent to the recipient.

External-body Contains a pointer to an object that exists elsewhere.

Image jpeg The image is in JPEG format, JFIF encoding.

gif The image is in GIF format.

Video mpeg MPEG format.

Audio Basic Single-channel 8-bit ISDN mu-law encoding at a sample rate of 8 kHz.

Application PostScript Adobe Postscript format.

octet-stream General binary data consisting of 8-bit bytes.

19.2 / S/MiMe 603

or text part could be accompanied by a voice commentary that is played while the picture or text is displayed.

For the multipart/alternative subtype, the various parts are different represen-tations of the same information. The following is an example:

From: Nathaniel Borenstein <[email protected]>

To: Ned Freed <[email protected]>

Subject: Formatted text mail

MIME-Version: 1.0

Content-Type: multipart/alternative;

boundary = boundary42

—boundary42

Content-Type: text/plain; charset = us-ascii. . . plain text version of message goes here . . ..

—boundary42

Content-Type: text/enriched

. . . RFC 1896 text/enriched version of same message goes here . . .

—boundary42—

In this subtype, the body parts are ordered in terms of increasing preference. For this example, if the recipient system is capable of displaying the message in the text/enriched format, this is done; otherwise, the plain text format is used.

The multipart/digest subtype is used when each of the body parts is inter-preted as an RFC 5322 message with headers. This subtype enables the construction of a message whose parts are individual messages. For example, the moderator of a group might collect e-mail messages from participants, bundle these messages, and send them out in one encapsulating MIME message.

The message type provides a number of important capabilities in MIME. The message/rfc822 subtype indicates that the body is an entire message, including header and body. Despite the name of this subtype, the encapsulated message may be not only a simple RFC 5322 message but also any MIME message.

The message/partial subtype enables fragmentation of a large message into a number of parts, which must be reassembled at the destination. For this subtype, three parameters are specified in the Content-Type: Message/Partial field: an id common to all fragments of the same message, a sequence number unique to each fragment, and the total number of fragments.

The message/external-body subtype indicates that the actual data to be con-veyed in this message are not contained in the body. Instead, the body contains the information needed to access the data. As with the other message types, the mes-sage/external-body subtype has an outer header and an encapsulated message with its own header. The only necessary field in the outer header is the Content-Type field, which identifies this as a message/external-body subtype. The inner header is the message header for the encapsulated message. The Content-Type field in the


outer header must include an access-type parameter, which indicates the method of access, such as FTP (file transfer protocol).

The application type refers to other kinds of data, typically either uninter-preted binary data or information to be processed by a mail-based application.

mime trAnsfer encodings The other major component of the MIME specifica-tion, in addition to content type specification, is a definition of transfer encodings for message bodies. The objective is to provide reliable delivery across the largest range of environments.

The MIME standard defines two methods of encoding data. The Content-Transfer-Encoding field can actually take on six values, as listed in Table 19.3. However, three of these values (7bit, 8bit, and binary) indicate that no encoding has been done but provide some information about the nature of the data. For SMTP transfer, it is safe to use the 7bit form. The 8bit and binary forms may be us-able in other mail transport contexts. Another Content-Transfer-Encoding value is x-token, which indicates that some other encoding scheme is used for which a name is to be supplied. This could be a vendor-specific or application-specific scheme. The two actual encoding schemes defined are quoted-printable and base64. Two schemes are defined to provide a choice between a transfer technique that is es-sentially human readable and one that is safe for all types of data in a way that is reasonably compact.

The quoted-printable transfer encoding is useful when the data consists largely of octets that correspond to printable ASCII characters. In essence, it represents nonsafe characters by the hexadecimal representation of their code and introduces reversible (soft) line breaks to limit message lines to 76 characters.

The base64 transfer encoding, also known as radix-64 encoding, is a common one for encoding arbitrary binary data in such a way as to be invulnerable to the processing by mail-transport programs. It is also used in PGP and is described in Appendix 19A.

A multipArt exAmple Figure 19.3, taken from RFC 2045, is the outline of a com-plex multipart message. The message has five parts to be displayed serially: two introductory plain text parts, an embedded multipart message, a richtext part, and

Table 19.3 MIME Transfer Encodings

7bit The data are all represented by short lines of ASCII characters.

8bit The lines are short, but there may be non-ASCII characters (octets with the high-order bit set).

binary Not only may non-ASCII characters be present, but the lines are not neces-sarily short enough for SMTP transport.

quoted-printable Encodes the data in such a way that if the data being encoded are mostly ASCII text, the encoded form of the data remains largely recognizable by humans.

base64 Encodes data by mapping 6-bit blocks of input to 8-bit blocks of output, all of which are printable ASCII characters.

x-token A named nonstandard encoding.

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MIME-Version: 1.0From: Nathaniel Borenstein <[email protected]>To: Ned Freed <[email protected]>Subject: A multipart exampleContent-Type: multipart/mixed;

boundary=unique-boundary-1

This is the preamble area of a multipart message. Mail readers that understand multipart format should ignorethis preamble. If you are reading this text, you might want to consider changing to a mail reader that understandshow to properly display multipart messages.

--unique-boundary-1

...Some text appears here...[Note that the preceding blank line means no header fields were given and this is text, with charset US ASCII.It could have been done with explicit typing as in the next part.]

--unique-boundary-1Content-type: text/plain; charset=US-ASCII

This could have been part of the previous part, but illustrates explicit versus implicit typing of body parts.

--unique-boundary-1Content-Type: multipart/parallel; boundary=unique-boundary-2

--unique-boundary-2Content-Type: audio/basicContent-Transfer-Encoding: base64

... base64-encoded 8000 Hz single-channel mu-law-format audio data goes here....

--unique-boundary-2Content-Type: image/jpegContent-Transfer-Encoding: base64

... base64-encoded image data goes here....

--unique-boundary-2--

--unique-boundary-1Content-type: text/enriched

This is <bold><italic>richtext.</italic></bold> <smaller>as defined in RFC 1896</smaller>

Isn't it <bigger><bigger>cool?</bigger></bigger>

--unique-boundary-1Content-Type: message/rfc822

From: (mailbox in US-ASCII)To: (address in US-ASCII)Subject: (subject in US-ASCII)Content-Type: Text/plain; charset=ISO-8859-1Content-Transfer-Encoding: Quoted-printable

... Additional text in ISO-8859-1 goes here ...

--unique-boundary-1--

Figure 19.3 Example MIME Message Structure


Table 19.4 Native and Canonical Form

Native Form

The body to be transmitted is created in the system’s native format. The native character set is used and, where appropriate, local end-of-line con-ventions are used as well. The body may be a UNIX-style text file, or a Sun raster image, or a VMS indexed file, or audio data in a system-dependent format stored only in memory, or anything else that corresponds to the local model for the representation of some form of information. Fundamentally, the data is created in the “native” form that corresponds to the type speci-fied by the media type.

Canonical Form

The entire body, including “out-of-band” information such as record lengths and possibly file attribute information, is converted to a universal canonical form. The specific media type of the body as well as its associated attributes dictate the nature of the canonical form that is used. Conversion to the proper canonical form may involve character set conversion, transformation of audio data, compression, or various other operations specific to the various media types. If character set conversion is involved, however, care must be taken to understand the semantics of the media type, which may have strong implications for any character set conversion (e.g., with regard to syntactically meaningful characters in a text subtype other than “plain”).

a closing encapsulated text message in a non-ASCII character set. The embedded multipart message has two parts to be displayed in parallel: a picture and an audio fragment.

cAnonicAl form An important concept in MIME and S/MIME is that of canonical form. Canonical form is a format, appropriate to the content type, that is standard-ized for use between systems. This is in contrast to native form, which is a format that may be peculiar to a particular system. Table 19.4, from RFC 2049, should help clarify this matter.

S/MIME Functionality

In terms of general functionality, S/MIME is very similar to PGP. Both offer the ability to sign and/or encrypt messages. In this subsection, we briefly summarize S/MIME capability. We then look in more detail at this capability by examining message formats and message preparation.

functions S/MIME provides the following functions.

• Enveloped data: This consists of encrypted content of any type and encrypted-content encryption keys for one or more recipients.

• Signed data: A digital signature is formed by taking the message digest of the content to be signed and then encrypting that with the private key of the signer. The content plus signature are then encoded using base64 encod-ing. A signed data message can only be viewed by a recipient with S/MIME capability.

• Clear-signed data: As with signed data, a digital signature of the content is formed. However, in this case, only the digital signature is encoded using

19.2 / S/MiMe 607

base64. As a result, recipients without S/MIME capability can view the mes-sage content, although they cannot verify the signature.

• Signed and enveloped data: Signed-only and encrypted-only entities may be nested, so that encrypted data may be signed and signed data or clear-signed data may be encrypted.

cryptogrAphic Algorithms Table 19.5 summarizes the cryptographic algorithms used in S/MIME. S/MIME uses the following terminology taken from RFC 2119 (Key Words for use in RFCs to Indicate Requirement Levels) to specify the require-ment level:

• MUST: The definition is an absolute requirement of the specification. An im-plementation must include this feature or function to be in conformance with the specification.

• SHOULD: There may exist valid reasons in particular circumstances to ignore this feature or function, but it is recommended that an implementation include the feature or function.

S/MIME incorporates three public-key algorithms. The Digital Signature Standard (DSS) described in Chapter 13 is the preferred algorithm for digital sig-nature. S/MIME lists Diffie-Hellman as the preferred algorithm for encrypting session keys; in fact, S/MIME uses a variant of Diffie-Hellman that does provide encryption/decryption, known as ElGamal (Chapter 10). As an alternative, RSA, described in Chapter 9, can be used for both signatures and session key encryption. These are the same algorithms used in PGP and provide a high level of security. For the hash function used to create the digital signature, the specification requires the 160-bit SHA-1 but recommends receiver support for the 128-bit MD5 for backward

Table 19.5 Cryptographic Algorithms Used in S/MIME

Function Requirement

Create a message digest to be used in forming a digital signature.

MUST support SHA-1.

Receiver SHOULD support MD5 for backward compatibility.

Encrypt message digest to form a digital signature.

Sending and receiving agents MUST support DSS.

Sending agents SHOULD support RSA encryption.

Receiving agents SHOULD support verification of RSA signa-tures with key sizes 512 bits to 1024 bits.

Encrypt session key for transmission with a message.

Sending and receiving agents SHOULD support Diffie-Hellman.

Sending and receiving agents MUST support RSA encryption with key sizes 512 bits to 1024 bits.

Encrypt message for transmission with a one-time session key.

Sending and receiving agents MUST support encryption with tripleDES.

Sending agents SHOULD support encryption with AES.

Sending agents SHOULD support encryption with RC2/40.

Create a message authentication code. Receiving agents MUST support HMAC with SHA-1.

Sending agents SHOULD support HMAC with SHA-1.


compatibility with older versions of S/MIME. As we discussed in Chapter 11, there is justifiable concern about the security of MD5, so SHA-1 is clearly the preferred alternative.

For message encryption, three-key triple DES (tripleDES) is recommended, but compliant implementations must support 40-bit RC2. The latter is a weak en-cryption algorithm but allows compliance with U.S. export controls.

The S/MIME specification includes a discussion of the procedure for decid-ing which content encryption algorithm to use. In essence, a sending agent has two decisions to make. First, the sending agent must determine if the receiving agent is capable of decrypting using a given encryption algorithm. Second, if the receiving agent is only capable of accepting weakly encrypted content, the sending agent must decide if it is acceptable to send using weak encryption. To support this decision process, a sending agent may announce its decrypting capabilities in order of prefer-ence for any message that it sends out. A receiving agent may store that information for future use.

The following rules, in the following order, should be followed by a sending agent.

1. If the sending agent has a list of preferred decrypting capabilities from an in-tended recipient, it SHOULD choose the first (highest preference) capability on the list that it is capable of using.

2. If the sending agent has no such list of capabilities from an intended re-cipient but has received one or more messages from the recipient, then the outgoing message SHOULD use the same encryption algorithm as was used on the last signed and encrypted message received from that intended recipient.

3. If the sending agent has no knowledge about the decryption capabilities of the intended recipient and is willing to risk that the recipient may not be able to decrypt the message, then the sending agent SHOULD use triple DES.

4. If the sending agent has no knowledge about the decryption capabilities of the intended recipient and is not willing to risk that the recipient may not be able to decrypt the message, then the sending agent MUST use RC2/40.

If a message is to be sent to multiple recipients and a common encryption algorithm cannot be selected for all, then the sending agent will need to send two messages. However, in that case, it is important to note that the security of the message is made vulnerable by the transmission of one copy with lower security.

S/MIME Messages

S/MIME makes use of a number of new MIME content types, which are shown in Table 19.6. All of the new application types use the designation PKCS. This refers to a set of public-key cryptography specifications issued by RSA Laboratories and made available for the S/MIME effort.

We examine each of these in turn after first looking at the general procedures for S/MIME message preparation.

19.2 / S/MiMe 609

securing A mime entity S/MIME secures a MIME entity with a signature, encryption, or both. A MIME entity may be an entire message (except for the RFC 5322 headers), or if the MIME content type is multipart, then a MIME entity is one or more of the subparts of the message. The MIME entity is prepared ac-cording to the normal rules for MIME message preparation. Then the MIME entity plus some security-related data, such as algorithm identifiers and certificates, are processed by S/MIME to produce what is known as a PKCS object. A PKCS object is then treated as message content and wrapped in MIME (provided with appropri-ate MIME headers). This process should become clear as we look at specific objects and provide examples.

In all cases, the message to be sent is converted to canonical form. In par-ticular, for a given type and subtype, the appropriate canonical form is used for the message content. For a multipart message, the appropriate canonical form is used for each subpart.

The use of transfer encoding requires special attention. For most cases, the result of applying the security algorithm will be to produce an object that is par-tially or totally represented in arbitrary binary data. This will then be wrapped in an outer MIME message, and transfer encoding can be applied at that point, typically base64. However, in the case of a multipart signed message (described in more de-tail later), the message content in one of the subparts is unchanged by the security process. Unless that content is 7bit, it should be transfer encoded using base64 or quoted-printable so that there is no danger of altering the content to which the sig-nature was applied.

We now look at each of the S/MIME content types.

envelopeddAtA An application/pkcs7-mime subtype is used for one of four categories of S/MIME processing, each with a unique smime-type parameter. In all cases, the resulting entity (referred to as an object) is represented in a form known as Basic Encoding Rules (BER), which is defined in ITU-T Recommendation X.209. The BER format consists of arbitrary octet strings and is therefore binary data. Such an object should be transfer encoded with base64 in the outer MIME message. We first look at envelopedData.

Table 19.6 S/MIME Content Types

Type Subtype smime Parameter Description

Multipart Signed A clear-signed message in two parts: one is the message and the other is the signature.

Application pkcs7-mime signedData A signed S/MIME entity.

pkcs7-mime envelopedData An encrypted S/MIME entity.

pkcs7-mime degenerate signedData

An entity containing only public-key certificates.

pkcs7-mime CompressedData A compressed S/MIME entity.

pkcs7- signature

signedData The content type of the signature subpart of a multipart/signed message.


The steps for preparing an envelopedData MIME entity are

1. Generate a pseudorandom session key for a particular symmetric encryption algorithm (RC2/40 or triple DES).

2. For each recipient, encrypt the session key with the recipient’s public RSA key.

3. For each recipient, prepare a block known as RecipientInfo that contains an identifier of the recipient’s public-key certificate,2 an identifier of the algo-rithm used to encrypt the session key, and the encrypted session key.

4. Encrypt the message content with the session key.

The RecipientInfo blocks followed by the encrypted content constitute the envelopedData. This information is then encoded into base64. A sample mes-sage (excluding the RFC 5322 headers) is

Content-Type: application/pkcs7-mime; smime-type=envelopeddata; name=smime.p7mContent-Transfer-Encoding: base64

Content-Disposition: attachment; filename=smime.p7m

rfvbnj756tbBghyHhHUujhJhjH77n8HHGT9HG4VQpfyF467GhIGfHfYT6

7n8HHGghyHhHUujhJh4VQpfyF467GhIGfHfYGTrfvbnjT6jH7756tbB9H

f8HHGTrfvhJhjH776tbB9HG4VQbnj7567GhIGfHfYT6ghyHhHUujpfyF4

0GhIGfHfQbnj756YT64V

To recover the encrypted message, the recipient first strips off the base64 en-coding. Then the recipient’s private key is used to recover the session key. Finally, the message content is decrypted with the session key.

signeddAtA The signedData smime-type can be used with one or more signers. For clarity, we confine our description to the case of a single digital signature. The steps for preparing a signedData MIME entity are

1. Select a message digest algorithm (SHA or MD5).

2. Compute the message digest (hash function) of the content to be signed.

3. Encrypt the message digest with the signer’s private key.

4. Prepare a block known as SignerInfo that contains the signer’s public-key certificate, an identifier of the message digest algorithm, an identifier of the al-gorithm used to encrypt the message digest, and the encrypted message digest.

The signedData entity consists of a series of blocks, including a message digest algorithm identifier, the message being signed, and SignerInfo. The signedData entity may also include a set of public-key certificates sufficient to constitute a chain from a recognized root or top-level certification authority to the signer. This information is then encoded into base64. A sample message (excluding the RFC 5322 headers) is

2This is an X.509 certificate, discussed later in this section.

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Content-Type: application/pkcs7-mime; smime-type=signed-data; name=smime.p7m

Content-Transfer-Encoding: base64

Content-Disposition: attachment; filename=smime.p7m

567GhIGfHfYT6ghyHhHUujpfyF4f8HHGTrfvhJhjH776tbB9HG4VQbnj7

77n8HHGT9HG4VQpfyF467GhIGfHfYT6rfvbnj756tbBghyHhHUujhJhjH

HUujhJh4VQpfyF467GhIGfHfYGTrfvbnjT6jH7756tbB9H7n8HHGghyHh

6YT64V0GhIGfHfQbnj75

To recover the signed message and verify the signature, the recipient first strips off the base64 encoding. Then the signer’s public key is used to decrypt the message digest. The recipient independently computes the message digest and com-pares it to the decrypted message digest to verify the signature.

cleAr signing Clear signing is achieved using the multipart content type with a signed subtype. As was mentioned, this signing process does not involve transforming the message to be signed, so that the message is sent “in the clear.” Thus, recipients with MIME capability but not S/MIME capability are able to read the incoming message.

A multipart/signed message has two parts. The first part can be any MIME type but must be prepared so that it will not be altered during transfer from source to destination. This means that if the first part is not 7bit, then it needs to be en-coded using base64 or quoted-printable. Then this part is processed in the same manner as signedData, but in this case an object with signedData format is created that has an empty message content field. This object is a detached signature. It is then transfer encoded using base64 to become the second part of the multipart/signed message. This second part has a MIME content type of application and a subtype of pkcs7-signature. Here is a sample message:

Content-Type: multipart/signed;

protocol="application/pkcs7-signature";micalg=sha1; boundary=boundary42

—boundary42

Content-Type: text/plain

This is a clear-signed message.

—boundary42

Content-Type: application/pkcs7-signature; name=smime.p7sContent-Transfer-Encoding: base64

Content-Disposition: attachment; filename=smime.p7s

ghyHhHUujhJhjH77n8HHGTrfvbnj756tbB9HG4VQpfyF467GhIGfHfYT6

4VQpfyF467GhIGfHfYT6jH77n8HHGghyHhHUujhJh756tbB9HGTrfvbnj

n8HHGTrfvhJhjH776tbB9HG4VQbnj7567GhIGfHfYT6ghyHhHUujpfyF4

7GhIGfHfYT64VQbnj756

—boundary42—


The protocol parameter indicates that this is a two-part clear-signed entity. The micalg parameter indicates the type of message digest used. The receiver can verify the signature by taking the message digest of the first part and comparing this to the message digest recovered from the signature in the second part.

registrAtion request Typically, an application or user will apply to a certification authority for a public-key certificate. The application/pkcs10 S/MIME entity is used to transfer a certification request. The certification request includes certifica-tion RequestInfo block, followed by an identifier of the public-key encryption algorithm, followed by the signature of the certificationRequestInfo block made using the sender’s private key. The certificationRequestInfo block includes a name of the certificate subject (the entity whose public key is to be certi-fied) and a bit-string representation of the user’s public key.

certificAtes-only messAge A message containing only certificates or a certificate revocation list (CRL) can be sent in response to a registration request. The message is an application/pkcs7-mime type/subtype with an smime-type parameter of degen-erate. The steps involved are the same as those for creating a signedData mes-sage, except that there is no message content and the signerInfo field is empty.

S/MIME Certificate Processing

S/MIME uses public-key certificates that conform to version 3 of X.509 (see Chapter 14). The key-management scheme used by S/MIME is in some ways a hy-brid between a strict X.509 certification hierarchy and PGP’s web of trust. As with the PGP model, S/MIME managers and/or users must configure each client with a list of trusted keys and with certificate revocation lists. That is, the responsibility is local for maintaining the certificates needed to verify incoming signatures and to encrypt outgoing messages. On the other hand, the certificates are signed by certi-fication authorities.

user Agent role An S/MIME user has several key-management functions to perform.

• Key generation: The user of some related administrative utility (e.g., one as-sociated with LAN management) MUST be capable of generating separate Diffie-Hellman and DSS key pairs and SHOULD be capable of generating RSA key pairs. Each key pair MUST be generated from a good source of nondeterministic random input and be protected in a secure fashion. A user agent SHOULD generate RSA key pairs with a length in the range of 768 to 1024 bits and MUST NOT generate a length of less than 512 bits.

• Registration: A user’s public key must be registered with a certification au-thority in order to receive an X.509 public-key certificate.

• Certificate storage and retrieval: A user requires access to a local list of certifi-cates in order to verify incoming signatures and to encrypt outgoing messages. Such a list could be maintained by the user or by some local administrative entity on behalf of a number of users.

19.2 / S/MiMe 613

verisign certificAtes There are several companies that provide certification au-thority (CA) services. For example, Nortel has designed an enterprise CA solu-tion and can provide S/MIME support within an organization. There are a number of Internet-based CAs, including VeriSign, GTE, and the U.S. Postal Service. Of these, the most widely used is the VeriSign CA service, a brief description of which we now provide.

VeriSign provides a CA service that is intended to be compatible with S/MIME and a variety of other applications. VeriSign issues X.509 certificates with the product name VeriSign Digital ID. As of early 1998, over 35,000 com-mercial Web sites were using VeriSign Server Digital IDs, and over a million consumer Digital IDs had been issued to users of Netscape and Microsoft browsers.

The information contained in a Digital ID depends on the type of Digital ID and its use. At a minimum, each Digital ID contains

• Owner’s public key

• Owner’s name or alias

• Expiration date of the Digital ID

• Serial number of the Digital ID

• Name of the certification authority that issued the Digital ID

• Digital signature of the certification authority that issued the Digital ID

Digital IDs can also contain other user-supplied information, including

• Address

• E-mail address

• Basic registration information (country, zip code, age, and gender)

VeriSign provides three levels, or classes, of security for public-key certifi-cates, as summarized in Table 19.7. A user requests a certificate online at VeriSign’s Web site or other participating Web sites. Class 1 and Class 2 requests are processed on line, and in most cases take only a few seconds to approve. Briefly, the following procedures are used.

• For Class 1 Digital IDs, VeriSign confirms the user’s e-mail address by send-ing a PIN and Digital ID pick-up information to the e-mail address provided in the application.

• For Class 2 Digital IDs, VeriSign verifies the information in the application through an automated comparison with a consumer database in addition to performing all of the checking associated with a Class 1 Digital ID. Finally, confirmation is sent to the specified postal address alerting the user that a Digital ID has been issued in his or her name.

• For Class 3 Digital IDs, VeriSign requires a higher level of identity assurance. An individual must prove his or her identity by providing notarized creden-tials or applying in person.


Enhanced Security Services

As of this writing, three enhanced security services have been proposed in an Internet draft. The details of these may change, and additional services may be added. The three services are

• Signed receipts: A signed receipt may be requested in a SignedData object. Returning a signed receipt provides proof of delivery to the originator of a message and allows the originator to demonstrate to a third party that the re-cipient received the message. In essence, the recipient signs the entire original message plus the original (sender’s) signature and appends the new signature to form a new S/MIME message.

• Security labels: A security label may be included in the authenticated attri-butes of a SignedData object. A security label is a set of security information regarding the sensitivity of the content that is protected by S/MIME encapsu-lation. The labels may be used for access control, by indicating which users are permitted access to an object. Other uses include priority (secret, confidential, restricted, and so on) or role based, describing which kind of people can see the information (e.g., patient’s health-care team, medical billing agents, etc.).

Table 19.7 VeriSign Public-Key Certificate Classes

Class 1 Class 2 Class 3

Summary of Confirmation of Identity

Automated unambig-uous name and e-mail address search.

Same as Class 1, plus auto-mated enrollment informa-tion check and automated address check.

Same as Class 1, plus personal presence and ID documents plus Class 2 automated ID check for individuals; business records (or filings) for organizations.

IA Private Key Protection

PCA: trustworthy hardware; CA: trust-worthy software or trustworthy hardware.

PCA and CA: trustworthy hardware.

PCA and CA: trustworthy hardware.

Certificate Applicant and Subscriber Private Key Protection

Encryption software (PIN protected) recommended but not required.

Encryption software (PIN protected) required.

Encryption software (PIN protected) required; hardware token recommended but not required.

Applications Implemented or Contemplated by Users

Web-browsing and certain e-mail usage.

Individual and intra- and inter-company e-mail, online subscriptions, password replacement, and software validation.

E-banking, corp. database access, personal banking, membership-based online services, content integrity services, e-commerce server, software validation; authenti-cation of LRAAs; and strong encryption for certain servers.

IA = Issuing AuthorityCA = Certification AuthorityPCA = VeriSign public primary certification authorityPIN = Personal Identification NumberLRAA = Local Registration Authority Administrator

19.3 / doMainkeyS identified Mail 615

• Secure mailing lists: When a user sends a message to multiple recipients, a cer-tain amount of per-recipient processing is required, including the use of each recipient’s public key. The user can be relieved of this work by employing the services of an S/MIME Mail List Agent (MLA). An MLA can take a single incoming message, perform the recipient-specific encryption for each recipi-ent, and forward the message. The originator of a message need only send the message to the MLA with encryption performed using the MLA’s public key.

19.3 dOMainkeys identified MaiL

DomainKeys Identified Mail (DKIM) is a specification for cryptographically signing e-mail messages, permitting a signing domain to claim responsibility for a message in the mail stream. Message recipients (or agents acting in their behalf) can verify the signature by querying the signer’s domain directly to retrieve the appropriate public key and thereby can confirm that the message was attested to by a party in possession of the private key for the signing domain. DKIM is a proposed Internet Standard (RFC 4871: DomainKeys Identified Mail (DKIM) Signatures). DKIM has been widely adopted by a range of e-mail providers, including corporations, govern-ment agencies, gmail, yahoo, and many Internet Service Providers (ISPs).

This section provides an overview of DKIM. Before beginning our discussion of DKIM, we introduce the standard Internet mail architecture. Then we look at the threat that DKIM is intended to address, and finally provide an overview of DKIM operation.

Internet Mail Architecture

To understand the operation of DKIM, it is useful to have a basic grasp of the Internet mail architecture, which is currently defined in RFC 5598. This subsection provides an overview of the basic concepts.

At its most fundamental level, the Internet mail architecture consists of a user world in the form of Message User Agents (MUA), and the transfer world, in the form of the Message Handling Service (MHS), which is composed of Message Transfer Agents (MTA). The MHS accepts a message from one user and delivers it to one or more other users, creating a virtual MUA-to-MUA exchange environ-ment. This architecture involves three types of interoperability. One is directly be-tween users: messages must be formatted by the MUA on behalf of the message author so that the message can be displayed to the message recipient by the destina-tion MUA. There are also interoperability requirements between the MUA and the MHS—first when a message is posted from an MUA to the MHS and later when it is delivered from the MHS to the destination MUA. Interoperability is required among the MTA components along the transfer path through the MHS.

Figure 19.4 illustrates the key components of the Internet mail architecture, which include the following.

• Message User Agent (MUA): Operates on behalf of user actors and user ap-plications. It is their representative within the e-mail service. Typically, this function is housed in the user’s computer and is referred to as a client e-mail


program or a local network e-mail server. The author MUA formats a mes-sage and performs initial submission into the MHS via a MSA. The recipient MUA processes received mail for storage and/or display to the recipient user.

• Mail Submission Agent (MSA): Accepts the message submitted by an MUA and enforces the policies of the hosting domain and the requirements of Internet standards. This function may be located together with the MUA or as a separate functional model. In the latter case, the Simple Mail Transfer Protocol (SMTP) is used between the MUA and the MSA.

• Message Transfer Agent (MTA): Relays mail for one application-level hop. It is like a packet switch or IP router in that its job is to make routing assess-ments and to move the message closer to the recipients. Relaying is performed by a sequence of MTAs until the message reaches a destination MDA. An MTA also adds trace information to the message header. SMTP is used be-tween MTAs and between an MTA and an MSA or MDA.

• Mail Delivery Agent (MDA): Responsible for transferring the message from the MHS to the MS.

• Message Store (MS): An MUA can employ a long-term MS. An MS can be located on a remote server or on the same machine as the MUA. Typically, an MUA retrieves messages from a remote server using POP (Post Office Protocol) or IMAP (Internet Message Access Protocol).

Two other concepts need to be defined. An administrative management domain (ADMD) is an Internet e-mail provider. Examples include a department

Message UserAgent (MUA)

Messageauthor

Messagerecipient

SMTP

SMTP

PTMSPTMS

(SMTP,local)

(SMTP,local)

(IMAP, POP,local)

Mail SubmissionAgent (MSA)

Message TransferAgent (MTA)


Message handlingsystem (MHS)


Mail DeliveryAgent (MDA)

Message Store(MS)

Message UserAgent (MUA)

Figure 19.4 Function Modules and Standardized Protocols Used Between Them


that operates a local mail relay (MTA), an IT department that operates an en-terprise mail relay, and an ISP that operates a public shared e-mail service. Each ADMD can have different operating policies and trust-based decision making. One obvious example is the distinction between mail that is exchanged within an organi-zation and mail that is exchanged between independent organizations. The rules for handling the two types of traffic tend to be quite different.

The Domain Name System (DNS) is a directory lookup service that provides a mapping between the name of a host on the Internet and its numerical address.

E-mail Threats

RFC 4686 (Analysis of Threats Motivating DomainKeys Identified Mail) describes the threats being addressed by DKIM in terms of the characteristics, capabilities, and location of potential attackers.

chArActeristics RFC 4686 characterizes the range of attackers on a spectrum of three levels of threat.

1. At the low end are attackers who simply want to send e-mail that a recipi-ent does not want to receive. The attacker can use one of a number of com-mercially available tools that allow the sender to falsify the origin address of messages. This makes it difficult for the receiver to filter spam on the basis of originating address or domain.

2. At the next level are professional senders of bulk spam mail. These attackers often operate as commercial enterprises and send messages on behalf of third parties. They employ more comprehensive tools for attack, including Mail Transfer Agents (MTAs) and registered domains and networks of compro-mised computers (zombies) to send messages and (in some cases) to harvest addresses to which to send.

3. The most sophisticated and financially motivated senders of messages are those who stand to receive substantial financial benefit, such as from an e-mail-based fraud scheme. These attackers can be expected to employ all of the above mechanisms and additionally may attack the Internet infrastructure itself, including DNS cache-poisoning attacks and IP routing attacks.

cApAbilities RFC 4686 lists the following as capabilities that an attacker might have.

1. Submit messages to MTAs and Message Submission Agents (MSAs) at mul-tiple locations in the Internet.

2. Construct arbitrary Message Header fields, including those claiming to be mailing lists, resenders, and other mail agents.

3. Sign messages on behalf of domains under their control.

4. Generate substantial numbers of either unsigned or apparently signed mes-sages that might be used to attempt a denial-of-service attack.

5. Resend messages that may have been previously signed by the domain.

6. Transmit messages using any envelope information desired.


7. Act as an authorized submitter for messages from a compromised computer.

8. Manipulation of IP routing. This could be used to submit messages from spe-cific IP addresses or difficult-to-trace addresses, or to cause diversion of mes-sages to a specific domain.

9. Limited influence over portions of DNS using mechanisms such as cache poi-soning. This might be used to influence message routing or to falsify advertise-ments of DNS-based keys or signing practices.

10. Access to significant computing resources, for example, through the conscrip-tion of worm-infected “zombie” computers. This could allow the “bad actor” to perform various types of brute-force attacks.

11. Ability to eavesdrop on existing traffic, perhaps from a wireless network.

locAtion DKIM focuses primarily on attackers located outside of the administra-tive units of the claimed originator and the recipient. These administrative units frequently correspond to the protected portions of the network adjacent to the orig-inator and recipient. It is in this area that the trust relationships required for authen-ticated message submission do not exist and do not scale adequately to be practical. Conversely, within these administrative units, there are other mechanisms (such as authenticated message submission) that are easier to deploy and more likely to be used than DKIM. External “bad actors” are usually attempting to exploit the “any-to-any” nature of e-mail that motivates most recipient MTAs to accept mes-sages from anywhere for delivery to their local domain. They may generate mes-sages without signatures, with incorrect signatures, or with correct signatures from domains with little traceability. They may also pose as mailing lists, greeting cards, or other agents that legitimately send or resend messages on behalf of others.

DKIM Strategy

DKIM is designed to provide an e-mail authentication technique that is transparent to the end user. In essence, a user’s e-mail message is signed by a private key of the administrative domain from which the e-mail originates. The signature covers all of the content of the message and some of the RFC 5322 message headers. At the re-ceiving end, the MDA can access the corresponding public key via a DNS and verify the signature, thus authenticating that the message comes from the claimed admin-istrative domain. Thus, mail that originates from somewhere else but claims to come from a given domain will not pass the authentication test and can be rejected. This approach differs from that of S/MIME and PGP, which use the originator’s private key to sign the content of the message. The motivation for DKIM is based on the following reasoning.3

1. S/MIME depends on both the sending and receiving users employing S/MIME. For almost all users, the bulk of incoming mail does not use S/MIME, and the bulk of the mail the user wants to send is to recipients not using S/MIME.

3The reasoning is expressed in terms of the use of S/MIME. The same argument applies to PGP.


2. S/MIME signs only the message content. Thus, RFC 5322 header information concerning origin can be compromised.

3. DKIM is not implemented in client programs (MUAs) and is therefore trans-parent to the user; the user need take no action.

4. DKIM applies to all mail from cooperating domains.

5. DKIM allows good senders to prove that they did send a particular message and to prevent forgers from masquerading as good senders.

Figure 19.5 is a simple example of the operation of DKIM. We begin with a message generated by a user and transmitted into the MHS to an MSA that is within the user’s administrative domain. An e-mail message is generated by an e-mail cli-ent program. The content of the message, plus selected RFC 5322 headers, is signed by the e-mail provider using the provider’s private key. The signer is associated with a domain, which could be a corporate local network, an ISP, or a public e-mail facility such as gmail. The signed message then passes through the Internet via a sequence of MTAs. At the destination, the MDA retrieves the public key for the incoming signature and verifies the signature before passing the message on to the destination e-mail client. The default signing algorithm is RSA with SHA-256. RSA with SHA-1 also may be used.

Mail originationnetwork

Mail deliverynetwork

DNS public-key query/response

DNS = domain name systemMDA = mail delivery agentMSA = mail submission agentMTA = message transfer agentMUA = message user agent

SMTP

MUA

MUA

SMTP

SMTP

Signer Verifier

SMTPPOP, IMAP

MT

AM

SA

MT

AM

DA

DN

S

Figure 19.5 Simple Example of DKIM Deployment


DKIM Functional Flow

Figure 19.6 provides a more detailed look at the elements of DKIM operation. Basic message processing is divided between a signing Administrative Management Domain (ADMD) and a verifying ADMD. At its simplest, this is between the originating ADMD and the delivering ADMD, but it can involve other ADMDs in the handling path.

Signing is performed by an authorized module within the signing ADMD and uses private information from a Key Store. Within the originating ADMD, this might be performed by the MUA, MSA, or an MTA. Verifying is performed by an authorized module within the verifying ADMD. Within a delivering ADMD, verifying might be performed by an MTA, MDA, or MUA. The module verifies the signature or determines whether a particular signature was required. Verifying the signature uses public information from the Key Store. If the signature passes,

Originating or Relaying ADMD:Sign Message with SDID

RFC 5322 Message

Yes

Pass Fail

No

Relaying or Delivering ADMDMessage signed?

Relaying or Delivering ADMD:Message signed?

Verifysignature

Privatekey

store

(paired)

Publickey

store

Remotesender

practices

Local infoon senderpractices

Reputation/accreditationinformation

Assessments

Messagefilteringengine

Checksigning

practices

Internet

Figure 19.6 DKIM Functional Flow


reputation information is used to assess the signer and that information is passed to the message filtering system. If the signature fails or there is no signature using the author’s domain, information about signing practices related to the author can be retrieved remotely and/or locally, and that information is passed to the message filtering system. For example, if the sender (e.g., gmail) uses DKIM but no DKIM signature is present, then the message may be considered fraudulent.

The signature is inserted into the RFC 5322 message as an additional header entry, starting with the keyword Dkim-Signature. You can view examples from your own incoming mail by using the View Long Headers (or similar wording) option for an incoming message. Here is an example:

Dkim-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=gamma; h=domainkey-signature:mime-version:received:date:message-id:subject :from:to:content-type:content-transfer-encoding;

bh=5mZvQDyCRuyLb1Y28K4zgS2MPOemFToDBgvbJ 7GO90s=;

b=PcUvPSDygb4ya5Dyj1rbZGp/VyRiScuaz7TTG J5qW5slM+klzv6kcfYdGDHzEVJW+Z

FetuPfF1ETOVhELtwH0zjSccOyPkEiblOf6gILO bm3DDRm3Ys1/FVrbhVOlA+/jH9Aei

uIIw/5iFnRbSH6qPDVv/beDQqAWQfA/wF7O5k=

Before a message is signed, a process known as canonicalization is performed on both the header and body of the RFC 5322 message. Canonicalization is neces-sary to deal with the possibility of minor changes in the message made en route, including character encoding, treatment of trailing white space in message lines, and the “folding” and “unfolding” of header lines. The intent of canonicalization is to make a minimal transformation of the message (for the purpose of signing; the mes-sage itself is not changed, so the canonicalization must be performed again by the verifier) that will give it its best chance of producing the same canonical value at the receiving end. DKIM defines two header canonicalization algorithms (“simple” and “relaxed”) and two for the body (with the same names). The simple algorithm tol-erates almost no modification, while the relaxed tolerates common modifications.

The signature includes a number of fields. Each field begins with a tag consist-ing of a tag code followed by an equals sign and ends with a semicolon. The fields include the following:

• v = DKIM version.

• a = Algorithm used to generate the signature; must be either rsa-sha1 or rsa-sha256.

• c = Canonicalization method used on the header and the body.

• d = A domain name used as an identifier to refer to the identity of a respon-sible person or organization. In DKIM, this identifier is called the Signing Domain IDentifier (SDID). In our example, this field indicates that the sender is using a gmail address.


• s = In order that different keys may be used in different circumstances for the same signing domain (allowing expiration of old keys, separate depart-mental signing, or the like), DKIM defines a selector (a name associated with a key), which is used by the verifier to retrieve the proper key during signature verification.

• h = Signed Header fields. A colon-separated list of header field names that identify the header fields presented to the signing algorithm. Note that in our example above, the signature covers the domainkey-signature field. This re-fers to an older algorithm (since replaced by DKIM) that is still in use.

• bh = The hash of the canonicalized body part of the message. This provides additional information for diagnosing signature verification failures.

• b = The signature data in base64 format; this is the encrypted hash code.


[LEIB07] provides an overview of DKIM.

LEIB07 Leiba, B., and Fenton, J. “DomainKeys Identified Mail (DKIM): Using Digital Signatures for Domain Verification.” Proceedings of Fourth Conference on E-mail and Anti-Spam (CEAS 07), 2007.

19.5 key terMs, review QuestiOns, and PrObLeMs

Key Terms

detached signatureDomainKeys Identified Mail

(DKIM)electronic mail

Multipurpose Internet Mail Extensions (MIME)

Pretty Good Privacy (PGP)radix 64

session keyS/MIMEtrustZIP

Review Questions 19.1 What are the five principal services provided by PGP? 19.2 What is the utility of a detached signature? 19.3 Why does PGP generate a signature before applying compression? 19.4 What is R64 conversion? 19.5 Why is R64 conversion useful for an e-mail application? 19.6 How does PGP use the concept of trust? 19.7 What is RFC 5322? 19.8 What is MIME? 19.9 What is S/MIME? 19.10 What is DKIM?

aPPendix 19a / radix-64 converSion 623

Problems 19.1 PGP makes use of the cipher feedback (CFB) mode of CAST-128, whereas most

symmetric encryption applications (other than key encryption) use the cipher block chaining (CBC) mode. We have

CBC: Ci = E(K, [Ci- 1 ⊕ Pi]); Pi = Ci- 1 ⊕ D(K, Ci)

CFB: Ci = Pi ⊕ E(K, Ci- 1); Pi = Ci ⊕ E(K, Ci- 1)

These two appear to provide equal security. Suggest a reason why PGP uses the CFB mode.

19.2 In the PGP scheme, what is the expected number of session keys generated before a previously created key is produced?

19.3 As discussed in Appendix P, a PGP user may have multiple public keys. So that a recipient knows which public key is being used by a sender, a key ID, consisting of the least significant 64 bits of the public key, is sent with the message. What is the prob-ability that a user with N public keys will have at least one duplicate key ID?

19.4 As discussed in Appendix P, the first 16 bits of the message digest in a PGP signature are translated in the clear. This enables the recipient to determine if the correct pub-lic key was used to decrypt the message digest by comparing the plaintext copy of the first two octets with the first two octets of the decrypted digest.

a. To what extent does this compromise the security of the hash algorithm? b. To what extent does it in fact perform its intended function, namely, to help deter-

mine if the correct RSA key was used to decrypt the digest? 19.5 For this problem and the next, consult Appendix P. In Figure P.2, each entry in the

public-key ring contains an Owner Trust field that indicates the degree of trust as-sociated with this public-key owner. Why is that not enough? That is, if this owner is trusted and this is supposed to be the owner’s public key, why is that trust not enough to permit PGP to use this public key?

19.6 What is the basic difference between X.509 and PGP in terms of key hierarchies and key trust?

19.7 Phil Zimmermann chose IDEA, three-key triple DES, and CAST-128 as symmetric encryption algorithms for PGP. Give reasons why each of the following symmetric encryption algorithms described in this book is suitable or unsuitable for PGP: DES, two-key triple DES, and AES.

19.8 Consider radix-64 conversion as a form of encryption. In this case, there is no key. But suppose that an opponent knew only that some form of substitution algorithm was being used to encrypt English text and did not guess that it was R64. How effec-tive would this algorithm be against cryptanalysis?

19.9 Encode the text “plaintext” using the following techniques. Assume characters are stored in 8-bit ASCII with zero parity.

a. Radix-64 b. Quoted-printable

aPPendix 19a radix-64 cOnversiOn

Both PGP and S/MIME make use of an encoding technique referred to as radix-64 conversion. This technique maps arbitrary binary input into printable character out-put. The form of encoding has the following relevant characteristics:

1. The range of the function is a character set that is universally representable at all sites, not a specific binary encoding of that character set. Thus, the characters themselves can be encoded into whatever form is needed by a


specific system. For example, the character “E” is represented in an ASCII-based system as hexadecimal 45 and in an EBCDIC-based system as hexa-decimal C5.

2. The character set consists of 65 printable characters, one of which is used for padding. With 26 = 64 available characters, each character can be used to rep-resent 6 bits of input.

3. No control characters are included in the set. Thus, a message encoded in radix 64 can traverse mail-handling systems that scan the data stream for con-trol characters.

4. The hyphen character “-” is not used. This character has significance in the RFC 5322 format and should therefore be avoided.

Table 19.8 shows the mapping of 6-bit input values to characters. The charac-ter set consists of the alphanumeric characters plus “+” and “/”. The “=” character is used as the padding character.

Figure 19.7 illustrates the simple mapping scheme. Binary input is processed in blocks of 3 octets (24 bits). Each set of 6 bits in the 24-bit block is mapped into a character. In the figure, the characters are shown encoded as 8-bit quantities. In this typical case, each 24-bit input is expanded to 32 bits of output.

Table 19.8 Radix-64 Encoding

6-bit Value

Character Encoding

6-bit Value

Character Encoding

6-bit Value

Character Encoding

6-bit Value

Character Encoding

0 A 16 Q 32 g 48 w

1 B 17 R 33 h 49 x

2 C 18 S 34 i 50 y

3 D 19 T 35 j 51 z

4 E 20 U 36 k 52 0

5 F 21 V 37 l 53 1

6 G 22 W 38 m 54 2

7 H 23 X 39 n 55 3

8 I 24 Y 40 o 56 4

9 J 25 Z 41 p 57 5

10 K 26 a 42 q 58 6

11 L 27 b 43 r 59 7

12 M 28 c 44 s 60 8

13 N 29 d 45 t 61 9

14 O 30 e 46 u 62 +

15 P 31 f 47 v 63 /

(pad) =

24 bits

R64 R64 R64 R64

4 characters = 32 bits

Figure 19.7 Printable Encoding of Binary Data into Radix-64 Format

For example, consider the 24-bit raw text sequence 00100011 01011100 10010001, which can be expressed in hexadecimal as 235C91. We arrange this input in blocks of 6 bits:

001000 110101 110010 010001

The extracted 6-bit decimal values are 8, 53, 50, and 17. Looking these up in Table 19.8 yields the radix-64 encoding as the following characters: I1yR. If these characters are stored in 8-bit ASCII format with parity bit set to zero, we have

01001001 00110001 01111001 01010010

In hexadecimal, this is 49317952. To summarize:

Input Data

Binary representation 00100011 01011100 10010001

Hexadecimal representation 235C91

Radix-64 Encoding of Input Data

Character representation I1yR

ASCII code (8 bit, zero parity) 01001001 00110001 01111001 01010010

Hexadecimal representation 49317952

aPPendix 19a / radix-64 converSion 625

IP SecurIty20.1 IP Security Overview

Applications of IPsec Benefits of IPsec Routing Applications IPsec Documents IPsec Services Transport and Tunnel Modes

20.2 IP Security Policy

Security Associations Security Association Database Security Policy Database IP Traffic Processing

20.3 Encapsulating Security Payload

ESP Format Encryption and Authentication Algorithms Padding Anti-Replay Service Transport and Tunnel Modes

20.4 Combining Security Associations

Authentication Plus Confidentiality Basic Combinations of Security Associations

20.5 Internet Key Exchange

Key Determination Protocol Header and Payload Formats

20.6 Cryptographic Suites



Chapter

626626

Chapter 20 / Ip SeCurIty 627

Learning Objectives


u Present an overview of IP security (IPsec).

u Explain the difference between transport mode and tunnel mode.

u Understand the concept of security association.

u Explain the difference between the security association database and the security policy database.

u Summarize the traffic processing functions performed by IPsec for outbound packets and for inbound packets.

u Present an overview of Encapsulating Security Payload.

u Discuss the alternatives for combining security associations.

u Present an overview of Internet Key Exchange.

u Summarize the alternative cryptographic suites approved for use with IPsec.

There are application-specific security mechanisms for a number of application areas, including electronic mail (S/MIME, PGP), client/server (Kerberos), Web ac-cess (Secure Sockets Layer), and others. However, users have security concerns that cut across protocol layers. For example, an enterprise can run a secure, private IP network by disallowing links to untrusted sites, encrypting packets that leave the premises, and authenticating packets that enter the premises. By implementing se-curity at the IP level, an organization can ensure secure networking not only for applications that have security mechanisms but also for the many security-ignorant applications.

IP-level security encompasses three functional areas: authentication, confiden-tiality, and key management. The authentication mechanism assures that a received packet was, in fact, transmitted by the party identified as the source in the packet header. In addition, this mechanism assures that the packet has not been altered in transit. The confidentiality facility enables communicating nodes to encrypt messages to prevent eavesdropping by third parties. The key management facility is concerned with the secure exchange of keys.

We begin this chapter with an overview of IP security (IPsec) and an introduc-tion to the IPsec architecture. We then look at each of the three functional areas in detail. Appendix L reviews Internet protocols.

If a secret piece of news is divulged by a spy before the time is ripe, he must be put to death, together with the man to whom the secret was told.


628 Chapter 20 / Ip SeCurIty

20.1 iP security Overview

In 1994, the Internet Architecture Board (IAB) issued a report titled “Security in the Internet Architecture” (RFC 1636). The report identified key areas for security mechanisms. Among these were the need to secure the network infrastructure from unauthorized monitoring and control of network traffic and the need to secure end-user-to-end-user traffic using authentication and encryption mechanisms.

To provide security, the IAB included authentication and encryption as nec-essary security features in the next-generation IP, which has been issued as IPv6. Fortunately, these security capabilities were designed to be usable both with the current IPv4 and the future IPv6. This means that vendors can begin offering these features now, and many vendors now do have some IPsec capability in their prod-ucts. The IPsec specification now exists as a set of Internet standards.

Applications of IPsec

IPsec provides the capability to secure communications across a LAN, across pri-vate and public WANs, and across the Internet. Examples of its use include:

• Secure branch office connectivity over the Internet: A company can build a secure virtual private network over the Internet or over a public WAN. This enables a business to rely heavily on the Internet and reduce its need for pri-vate networks, saving costs and network management overhead.

• Secure remote access over the Internet: An end user whose system is equipped with IP security protocols can make a local call to an Internet Service Provider (ISP) and gain secure access to a company network. This reduces the cost of toll charges for traveling employees and telecommuters.

• Establishing extranet and intranet connectivity with partners: IPsec can be used to secure communication with other organizations, ensuring authentica-tion and confidentiality and providing a key exchange mechanism.

• Enhancing electronic commerce security: Even though some Web and elec-tronic commerce applications have built-in security protocols, the use of IPsec enhances that security. IPsec guarantees that all traffic designated by the net-work administrator is both encrypted and authenticated, adding an additional layer of security to whatever is provided at the application layer.

The principal feature of IPsec that enables it to support these varied applica-tions is that it can encrypt and/or authenticate all traffic at the IP level. Thus, all distributed applications (including remote logon, client/server, e-mail, file transfer, Web access, and so on) can be secured.

Figure 20.1 is a typical scenario of IPsec usage. An organization maintains LANs at dispersed locations. Nonsecure IP traffic is conducted on each LAN. For traffic offsite, through some sort of private or public WAN, IPsec protocols are used. These protocols operate in networking devices, such as a router or firewall, that connect each LAN to the outside world. The IPsec networking device will typically encrypt and compress all traffic going into the WAN and decrypt and decompress traffic coming from the WAN; these operations are transparent to workstations and

20.1 / Ip SeCurIty OvervIew 629

servers on the LAN. Secure transmission is also possible with individual users who dial into the WAN. Such user workstations must implement the IPsec protocols to provide security.

Benefits of IPsec

Some of the benefits of IPsec:

• When IPsec is implemented in a firewall or router, it provides strong secu-rity that can be applied to all traffic crossing the perimeter. Traffic within a company or workgroup does not incur the overhead of security-related processing.

• IPsec in a firewall is resistant to bypass if all traffic from the outside must use IP and the firewall is the only means of entrance from the Internet into the organization.

• IPsec is below the transport layer (TCP, UDP) and so is transparent to appli-cations. There is no need to change software on a user or server system when IPsec is implemented in the firewall or router. Even if IPsec is implemented in end systems, upper-layer software, including applications, is not affected.

• IPsec can be transparent to end users. There is no need to train users on secu-rity mechanisms, issue keying material on a per-user basis, or revoke keying material when users leave the organization.

IPHeader

IPPayload

IPHeader

IPPayload

IPHeader

IPSecHeader

Secure IPPayload

Networking devicewith IPSec

Ethernetswitch

Ethernetswitch

User systemwith IPSec

Networking devicewith IPSec

Public (Internet)or PrivateNetwork

IPHea

der

IPSe

cHea

der

Secu

re IP

Payl

oad

IPHeader

IPSec

Header

Secure IP

Payload

Figure 20.1 An IP Security Scenario


• IPsec can provide security for individual users if needed. This is useful for off-site workers and for setting up a secure virtual subnetwork within an organiza-tion for sensitive applications.

Routing Applications

In addition to supporting end users and protecting premises systems and networks, IPsec can play a vital role in the routing architecture required for internetworking. [HUIT98] lists the following examples of the use of IPsec. IPsec can assure that

• A router advertisement (a new router advertises its presence) comes from an authorized router.

• A neighbor advertisement (a router seeks to establish or maintain a neighbor relationship with a router in another routing domain) comes from an autho-rized router.

• A redirect message comes from the router to which the initial IP packet was sent.

• A routing update is not forged.

Without such security measures, an opponent can disrupt communications or divert some traffic. Routing protocols such as Open Shortest Path First (OSPF) should be run on top of security associations between routers that are defined by IPsec.

IPsec Documents

IPsec encompasses three functional areas: authentication, confidentiality, and key management. The totality of the IPsec specification is scattered across dozens of RFCs and draft IETF documents, making this the most complex and difficult to grasp of all IETF specifications. The best way to grasp the scope of IPsec is to consult the latest version of the IPsec document roadmap, which as of this writing is RFC 6071 [IP Security (IPsec) and Internet Key Exchange (IKE) Document Roadmap, February 2011]. The documents can be categorized into the following groups.

• Architecture: Covers the general concepts, security requirements, definitions, and mechanisms defining IPsec technology. The current specification is RFC 4301, Security Architecture for the Internet Protocol.

• Authentication Header (AH): AH is an extension header to provide mes-sage authentication. The current specification is RFC 4302, IP Authentication Header. Because message authentication is provided by ESP, the use of AH is deprecated. It is included in IPsecv3 for backward compatibility but should not be used in new applications. We do not discuss AH in this chapter.

• Encapsulating Security Payload (ESP): ESP consists of an encapsulating header and trailer used to provide encryption or combined encryption/au-thentication. The current specification is RFC 4303, IP Encapsulating Security Payload (ESP).

• Internet Key Exchange (IKE): This is a collection of documents describing the key management schemes for use with IPsec. The main specification is RFC 5996, Internet Key Exchange (IKEv2) Protocol, but there are a number of related RFCs.

20.1 / Ip SeCurIty OvervIew 631

• Cryptographic algorithms: This category encompasses a large set of documents that define and describe cryptographic algorithms for encryption, message au-thentication, pseudorandom functions (PRFs), and cryptographic key exchange.

• Other: There are a variety of other IPsec-related RFCs, including those deal-ing with security policy and management information base (MIB) content.

IPsec Services

IPsec provides security services at the IP layer by enabling a system to select re-quired security protocols, determine the algorithm(s) to use for the service(s), and put in place any cryptographic keys required to provide the requested services. Two protocols are used to provide security: an authentication protocol designated by the header of the protocol, Authentication Header (AH); and a combined encryption/ authentication protocol designated by the format of the packet for that protocol, Encapsulating Security Payload (ESP). RFC 4301 lists the following services:

• Access control

• Connectionless integrity

• Data origin authentication

• Rejection of replayed packets (a form of partial sequence integrity)

• Confidentiality (encryption)

• Limited traffic flow confidentiality

Transport and Tunnel Modes

Both AH and ESP support two modes of use: transport and tunnel mode. The oper-ation of these two modes is best understood in the context of a description of ESP, which is covered in Section 20.3. Here we provide a brief overview.

TransporT Mode Transport mode provides protection primarily for upper-layer protocols. That is, transport mode protection extends to the payload of an IP packet.1 Examples include a TCP or UDP segment or an ICMP packet, all of which operate directly above IP in a host protocol stack. Typically, transport mode is used for end-to-end communication between two hosts (e.g., a client and a server, or two worksta-tions). When a host runs AH or ESP over IPv4, the payload is the data that normally follow the IP header. For IPv6, the payload is the data that normally follow both the IP header and any IPv6 extensions headers that are present, with the possible excep-tion of the destination options header, which may be included in the protection.

ESP in transport mode encrypts and optionally authenticates the IP payload but not the IP header. AH in transport mode authenticates the IP payload and se-lected portions of the IP header.

Tunnel Mode Tunnel mode provides protection to the entire IP packet. To achieve this, after the AH or ESP fields are added to the IP packet, the entire packet plus security fields is treated as the payload of new outer IP packet with a new outer IP

1In this chapter, the term IP packet refers to either an IPv4 datagram or an IPv6 packet.


header. The entire original, inner, packet travels through a tunnel from one point of an IP network to another; no routers along the way are able to examine the inner IP header. Because the original packet is encapsulated, the new, larger packet may have totally different source and destination addresses, adding to the security. Tunnel mode is used when one or both ends of a security association (SA) are a security gateway, such as a firewall or router that implements IPsec. With tunnel mode, a number of hosts on networks behind firewalls may engage in secure communica-tions without implementing IPsec. The unprotected packets generated by such hosts are tunneled through external networks by tunnel mode SAs set up by the IPsec software in the firewall or secure router at the boundary of the local network.

Here is an example of how tunnel mode IPsec operates. Host A on a network generates an IP packet with the destination address of host B on another network. This packet is routed from the originating host to a firewall or secure router at the boundary of A’s network. The firewall filters all outgoing packets to determine the need for IPsec processing. If this packet from A to B requires IPsec, the firewall performs IPsec processing and encapsulates the packet with an outer IP header. The source IP address of this outer IP packet is this firewall, and the destination address may be a firewall that forms the boundary to B’s local network. This packet is now routed to B’s firewall, with intermediate routers examining only the outer IP header. At B’s firewall, the outer IP header is stripped off, and the inner packet is delivered to B.

ESP in tunnel mode encrypts and optionally authenticates the entire inner IP packet, including the inner IP header. AH in tunnel mode authenticates the entire inner IP packet and selected portions of the outer IP header.

Table 20.1 summarizes transport and tunnel mode functionality.

20.2 iP security POLicy

Fundamental to the operation of IPsec is the concept of a security policy applied to each IP packet that transits from a source to a destination. IPsec policy is de-termined primarily by the interaction of two databases, the security association database (SAD) and the security policy database (SPD). This section provides an

Table 20.1 Tunnel Mode and Transport Mode Functionality

Transport Mode SA Tunnel Mode SA

AH Authenticates IP payload and selected portions of IP header and IPv6 extension headers.

Authenticates entire inner IP packet (inner header plus IP payload) plus selected portions of outer IP header and outer IPv6 extension headers.

ESP Encrypts IP payload and any IPv6 exten-sion headers following the ESP header.

Encrypts entire inner IP packet.

ESP with Authentication

Encrypts IP payload and any IPv6 extension headers following the ESP header. Authenticates IP payload but not IP header.

Encrypts entire inner IP packet. Authenticates inner IP packet.

20.2 / Ip SeCurIty pOlICy 633

overview of these two databases and then summarizes their use during IPsec opera-tion. Figure 20.2 illustrates the relevant relationships.

Security Associations

A key concept that appears in both the authentication and confidentiality mecha-nisms for IP is the security association (SA). An association is a one-way logical connection between a sender and a receiver that affords security services to the traf-fic carried on it. If a peer relationship is needed for two-way secure exchange, then two security associations are required.

A security association is uniquely identified by three parameters.

• Security Parameters Index (SPI): A 32-bit unsigned integer assigned to this SA and having local significance only. The SPI is carried in AH and ESP head-ers to enable the receiving system to select the SA under which a received packet will be processed.

• IP Destination Address: This is the address of the destination endpoint of the SA, which may be an end-user system or a network system such as a firewall or router.

• Security Protocol Identifier: This field from the outer IP header indicates whether the association is an AH or ESP security association.

Hence, in any IP packet, the security association is uniquely identified by the Destination Address in the IPv4 or IPv6 header and the SPI in the enclosed exten-sion header (AH or ESP).

Security Association Database

In each IPsec implementation, there is a nominal2 Security Association Database that defines the parameters associated with each SA. A security association is nor-mally defined by the following parameters in an SAD entry.

SPD SPD

SAD

IKEv2 IKEv2

IPsecv3 IPsecv3

Securityassociationdatabase

Key exchange

IKE SA

IPsec SA Pair

ESP protects data

Securityassociationdatabase

Securitypolicy

database

Securitypolicy

database

SAD

Figure 20.2 IPsec Architecture

2Nominal in the sense that the functionality provided by a Security Association Database must be present in any IPsec implementation, but the way in which that functionality is provided is up to the implementer.


• Security Parameter Index: A 32-bit value selected by the receiving end of an SA to uniquely identify the SA. In an SAD entry for an outbound SA, the SPI is used to construct the packet’s AH or ESP header. In an SAD entry for an inbound SA, the SPI is used to map traffic to the appropriate SA.

• Sequence Number Counter: A 32-bit value used to generate the Sequence Number field in AH or ESP headers, described in Section 20.3 (required for all implementations).

• Sequence Counter Overflow: A flag indicating whether overflow of the Sequence Number Counter should generate an auditable event and prevent further transmission of packets on this SA (required for all implementations).

• Anti-Replay Window: Used to determine whether an inbound AH or ESP packet is a replay, described in Section 20.3 (required for all implementations).

• AH Information: Authentication algorithm, keys, key lifetimes, and related parameters being used with AH (required for AH implementations).

• ESP Information: Encryption and authentication algorithm, keys, initializa-tion values, key lifetimes, and related parameters being used with ESP (required for ESP implementations).

• Lifetime of this Security Association: A time interval or byte count after which an SA must be replaced with a new SA (and new SPI) or terminated, plus an indication of which of these actions should occur (required for all implementations).

• IPsec Protocol Mode: Tunnel, transport, or wildcard.

• Path MTU: Any observed path maximum transmission unit (maximum size of a packet that can be transmitted without fragmentation) and aging variables (required for all implementations).

The key management mechanism that is used to distribute keys is coupled to the authentication and privacy mechanisms only by way of the Security Parameters Index (SPI). Hence, authentication and privacy have been specified independent of any specific key management mechanism.

IPsec provides the user with considerable flexibility in the way in which IPsec services are applied to IP traffic. As we will see later, SAs can be combined in a number of ways to yield the desired user configuration. Furthermore, IPsec pro-vides a high degree of granularity in discriminating between traffic that is afforded IPsec protection and traffic that is allowed to bypass IPsec, as in the former case relating IP traffic to specific SAs.

Security Policy Database

The means by which IP traffic is related to specific SAs (or no SA in the case of traffic allowed to bypass IPsec) is the nominal Security Policy Database (SPD). In its simplest form, an SPD contains entries, each of which defines a subset of IP traf-fic and points to an SA for that traffic. In more complex environments, there may be multiple entries that potentially relate to a single SA or multiple SAs associated with a single SPD entry. The reader is referred to the relevant IPsec documents for a full discussion.


Each SPD entry is defined by a set of IP and upper-layer protocol field values, called selectors. In effect, these selectors are used to filter outgoing traffic in order to map it into a particular SA. Outbound processing obeys the following general sequence for each IP packet.

1. Compare the values of the appropriate fields in the packet (the selector fields) against the SPD to find a matching SPD entry, which will point to zero or more SAs.

2. Determine the SA if any for this packet and its associated SPI.

3. Do the required IPsec processing (i.e., AH or ESP processing).

The following selectors determine an SPD entry:

• Remote IP Address: This may be a single IP address, an enumerated list or range of addresses, or a wildcard (mask) address. The latter two are required to support more than one destination system sharing the same SA (e.g., be-hind a firewall).

• Local IP Address: This may be a single IP address, an enumerated list or range of addresses, or a wildcard (mask) address. The latter two are required to sup-port more than one source system sharing the same SA (e.g., behind a firewall).

• Next Layer Protocol: The IP protocol header (IPv4, IPv6, or IPv6 Extension) includes a field (Protocol for IPv4, Next Header for IPv6 or IPv6 Extension) that designates the protocol operating over IP. This is an individual protocol number, ANY, or for IPv6 only, OPAQUE. If AH or ESP is used, then this IP protocol header immediately precedes the AH or ESP header in the packet.

• Name: A user identifier from the operating system. This is not a field in the IP or upper-layer headers but is available if IPsec is running on the same operat-ing system as the user.

• Local and Remote Ports: These may be individual TCP or UDP port values, an enumerated list of ports, or a wildcard port.

Table 20.2 provides an example of an SPD on a host system (as opposed to a network system such as a firewall or router). This table reflects the following

Table 20.2 Host SPD Example

Protocol Local IP Port Remote IP Port Action Comment

UDP 1.2.3.101 500 * 500 BYPASS IKE

ICMP 1.2.3.101 * * * BYPASS Error messages

* 1.2.3.101 * 1.2.3.0/24 * PROTECT: ESP intransport-mode

Encrypt intranet traffic

TCP 1.2.3.101 * 1.2.4.10 80 PROTECT: ESP intransport-mode

Encrypt to server

TCP 1.2.3.101 * 1.2.4.10 443 BYPASS TLS: avoid double encryption

* 1.2.3.101 * 1.2.4.0/24 * DISCARD Others in DMZ

* 1.2.3.101 * * * BYPASS Internet


configuration: A local network configuration consists of two networks. The basic corporate network configuration has the IP network number 1.2.3.0/24. The local configuration also includes a secure LAN, often known as a DMZ, that is identified as 1.2.4.0/24. The DMZ is protected from both the outside world and the rest of the corporate LAN by firewalls. The host in this example has the IP address 1.2.3.10, and it is authorized to connect to the server 1.2.4.10 in the DMZ.

The entries in the SPD should be self-explanatory. For example, UDP port 500 is the designated port for IKE. Any traffic from the local host to a remote host for purposes of an IKE exchange bypasses the IPsec processing.

IP Traffic Processing

IPsec is executed on a packet-by-packet basis. When IPsec is implemented, each outbound IP packet is processed by the IPsec logic before transmission, and each inbound packet is processed by the IPsec logic after reception and before passing the packet contents on to the next higher layer (e.g., TCP or UDP). We look at the logic of these two situations in turn.

ouTbound packeTs Figure 20.3 highlights the main elements of IPsec processing for outbound traffic. A block of data from a higher layer, such as TCP, is passed

Searchsecurity policy

database

Searchsecurity association

database

Determinepolicy

Outbound IP packet(e.g., from TCP or UDP)

Discardpacket

No matchfound

No matchfound

Match found

Matchfound

DISCARD PROTECT

BYPASS

Forwardpacket via

IP

Internetkey

exchange

Process(AH/ESP)

Figure 20.3 Processing Model for Outbound Packets


down to the IP layer and an IP packet is formed, consisting of an IP header and an IP body. Then the following steps occur:

1. IPsec searches the SPD for a match to this packet.

2. If no match is found, then the packet is discarded and an error message is generated.

3. If a match is found, further processing is determined by the first matching entry in the SPD. If the policy for this packet is DISCARD, then the packet is discarded. If the policy is BYPASS, then there is no further IPsec processing; the packet is forwarded to the network for transmission.

4. If the policy is PROTECT, then a search is made of the SAD for a matching entry. If no entry is found, then IKE is invoked to create an SA with the ap-propriate keys and an entry is made in the SA.

5. The matching entry in the SAD determines the processing for this packet. Either encryption, authentication, or both can be performed, and either trans-port or tunnel mode can be used. The packet is then forwarded to the network for transmission.

Inbound packeTs Figure 20.4 highlights the main elements of IPsec processing for inbound traffic. An incoming IP packet triggers the IPsec processing. The following steps occur:

1. IPsec determines whether this is an unsecured IP packet or one that has ESP or AH headers/trailers, by examining the IP Protocol field (IPv4) or Next Header field (IPv6).

Searchsecurity policy

database

Searchsecurity association

database

Packettype

Inbound IP packet(from Internet)

Discardpacket

No matchfound

cesPIPI

NotBYPASS

MatchfoundBYPASS

Deliver packetto higher layer

(e.g., TCP, UDP)

Process(AH/ESP)

Figure 20.4 Processing Model for Inbound Packets


2. If the packet is unsecured, IPsec searches the SPD for a match to this packet. If the first matching entry has a policy of BYPASS, the IP header is processed and stripped off and the packet body is delivered to the next higher layer, such as TCP. If the first matching entry has a policy of PROTECT or DISCARD, or if there is no matching entry, the packet is discarded.

3. For a secured packet, IPsec searches the SAD. If no match is found, the packet is discarded. Otherwise, IPsec applies the appropriate ESP or AH processing. Then, the IP header is processed and stripped off and the packet body is deliv-ered to the next higher layer, such as TCP.

20.3 encaPsuLating security PayLOad

ESP can be used to provide confidentiality, data origin authentication, connection-less integrity, an anti-replay service (a form of partial sequence integrity), and (lim-ited) traffic flow confidentiality. The set of services provided depends on options selected at the time of Security Association (SA) establishment and on the location of the implementation in a network topology.

ESP can work with a variety of encryption and authentication algorithms, in-cluding authenticated encryption algorithms such as GCM.

ESP Format

Figure 20.5a shows the top-level format of an ESP packet. It contains the following fields.

• Security Parameters Index (32 bits): Identifies a security association.

• Sequence Number (32 bits): A monotonically increasing counter value; this provides an anti-replay function, as discussed for AH.

• Payload Data (variable): This is a transport-level segment (transport mode) or IP packet (tunnel mode) that is protected by encryption.

• Padding (0–255 bytes): The purpose of this field is discussed later.

• Pad Length (8 bits): Indicates the number of pad bytes immediately preceding this field.

• Next Header (8 bits): Identifies the type of data contained in the payload data field by identifying the first header in that payload (e.g., an extension header in IPv6, or an upper-layer protocol such as TCP).

• Integrity Check Value (variable): A variable-length field (must be an integral number of 32-bit words) that contains the Integrity Check Value computed over the ESP packet minus the Authentication Data field.

When any combined mode algorithm is employed, the algorithm itself is ex-pected to return both decrypted plaintext and a pass/fail indication for the integrity check. For combined mode algorithms, the ICV that would normally appear at the end of the ESP packet (when integrity is selected) may be omitted. When the ICV is omitted and integrity is selected, it is the responsibility of the combined mode

20.3 / enCapSulatIng SeCurIty paylOad 639

algorithm to encode within the Payload Data an ICV-equivalent means of verifying the integrity of the packet.

Two additional fields may be present in the payload (Figure 20.5b). An initial-ization value (IV), or nonce, is present if this is required by the encryption or authenticated encryption algorithm used for ESP. If tunnel mode is being used, then the IPsec implementation may add traffic flow confidentiality (TFC) padding after the Payload Data and before the Padding field, as explained subsequently.

Encryption and Authentication Algorithms

The Payload Data, Padding, Pad Length, and Next Header fields are encrypted by the ESP service. If the algorithm used to encrypt the payload requires cryptographic synchronization data, such as an initialization vector (IV), then these data may be carried explicitly at the beginning of the Payload Data field. If included, an IV is usually not encrypted, although it is often referred to as being part of the ciphertext.

Security parameters index (SPI)

32 bits

Sequence number

Padding (0–255 bytes)Pad length Next header

Payload data (variable)

Integrity check value - ICV (variable)

ICV

cov

erag

e

Enc

rypt

edE

ncry

pted

(a) Top-level format of an ESP Packet

(b) Substructure of payload data

Security parameters index (SPI)

Sequence number

Initialization value - IV (optional)

Padding (0–255 bytes)

TFC padding (optional, variable)

Pad length Next header

Rest of payload data (variable)

Integrity check value - ICV (variable)

ICV

cov

erag

e

Pay

load

Figure 20.5 ESP Packet Format


The ICV field is optional. It is present only if the integrity service is selected and is provided by either a separate integrity algorithm or a combined mode algo-rithm that uses an ICV. The ICV is computed after the encryption is performed. This order of processing facilitates rapid detection and rejection of replayed or bogus packets by the receiver prior to decrypting the packet, hence potentially reducing the impact of denial of service (DoS) attacks. It also allows for the pos-sibility of parallel processing of packets at the receiver that is decryption can take place in parallel with integrity checking. Note that because the ICV is not pro-tected by encryption, a keyed integrity algorithm must be employed to compute the ICV.

Padding

The Padding field serves several purposes:

• If an encryption algorithm requires the plaintext to be a multiple of some number of bytes (e.g., the multiple of a single block for a block cipher), the Padding field is used to expand the plaintext (consisting of the Payload Data, Padding, Pad Length, and Next Header fields) to the required length.

• The ESP format requires that the Pad Length and Next Header fields be right aligned within a 32-bit word. Equivalently, the ciphertext must be an integer multiple of 32 bits. The Padding field is used to assure this alignment.

• Additional padding may be added to provide partial traffic-flow confidential-ity by concealing the actual length of the payload.

Anti-Replay Service

A replay attack is one in which an attacker obtains a copy of an authenticated packet and later transmits it to the intended destination. The receipt of duplicate, authenticated IP packets may disrupt service in some way or may have some other undesired consequence. The Sequence Number field is designed to thwart such at-tacks. First, we discuss sequence number generation by the sender, and then we look at how it is processed by the recipient.

When a new SA is established, the sender initializes a sequence number counter to 0. Each time that a packet is sent on this SA, the sender increments the counter and places the value in the Sequence Number field. Thus, the first value to be used is 1. If anti-replay is enabled (the default), the sender must not allow the sequence number to cycle past 232 - 1 back to zero. Otherwise, there would be multiple valid packets with the same sequence number. If the limit of 232 - 1 is reached, the sender should terminate this SA and negotiate a new SA with a new key.

Because IP is a connectionless, unreliable service, the protocol does not guarantee that packets will be delivered in order and does not guarantee that all packets will be delivered. Therefore, the IPsec authentication document dictates that the receiver should implement a window of size W, with a default of W = 64. The right edge of the window represents the highest sequence number, N, so far received for a valid packet. For any packet with a sequence number in the range from N - W + 1 to N that has been correctly received (i.e., properly authenticated), the


corresponding slot in the window is marked (Figure 20.6). Inbound processing pro-ceeds as follows when a packet is received:

1. If the received packet falls within the window and is new, the MAC is checked. If the packet is authenticated, the corresponding slot in the window is marked.

2. If the received packet is to the right of the window and is new, the MAC is checked. If the packet is authenticated, the window is advanced so that this sequence number is the right edge of the window, and the corresponding slot in the window is marked.

3. If the received packet is to the left of the window or if authentication fails, the packet is discarded; this is an auditable event.

Transport and Tunnel Modes

Figure 20.7 shows two ways in which the IPsec ESP service can be used. In the upper part of the figure, encryption (and optionally authentication) is provided di-rectly between two hosts. Figure 20.7b shows how tunnel mode operation can be used to set up a virtual private network. In this example, an organization has four private networks interconnected across the Internet. Hosts on the internal networks use the Internet for transport of data but do not interact with other Internet-based hosts. By terminating the tunnels at the security gateway to each internal network, the configuration allows the hosts to avoid implementing the security capability. The former technique is supported by a transport mode SA, while the latter tech-nique uses a tunnel mode SA.

In this section, we look at the scope of ESP for the two modes. The consid-erations are somewhat different for IPv4 and IPv6. We use the packet formats of Figure 20.8a as a starting point.

TransporT Mode esp Transport mode ESP is used to encrypt and optionally au-thenticate the data carried by IP (e.g., a TCP segment), as shown in Figure 20.8b. For this mode using IPv4, the ESP header is inserted into the IP packet immedi-ately prior to the transport-layer header (e.g., TCP, UDP, ICMP), and an ESP

Fixed window size W

N

N � 1N � W

Marked if validpacket received

Unmarked if validpacket not yet received

• • •

Advance window ifvalid packet to theright is received

Figure 20.6 Anti-replay Mechanism


trailer (Padding, Pad Length, and Next Header fields) is placed after the IP packet. If authentication is selected, the ESP Authentication Data field is added after the ESP trailer. The entire transport-level segment plus the ESP trailer are encrypted. Authentication covers all of the ciphertext plus the ESP header.

In the context of IPv6, ESP is viewed as an end-to-end payload; that is, it is not examined or processed by intermediate routers. Therefore, the ESP header ap-pears after the IPv6 base header and the hop-by-hop, routing, and fragment exten-sion headers. The destination options extension header could appear before or after the ESP header, depending on the semantics desired. For IPv6, encryption covers the entire transport-level segment plus the ESP trailer plus the destination options extension header if it occurs after the ESP header. Again, authentication covers the ciphertext plus the ESP header.

Transport mode operation may be summarized as follows.

1. At the source, the block of data consisting of the ESP trailer plus the entire transport-layer segment is encrypted and the plaintext of this block is replaced with its ciphertext to form the IP packet for transmission. Authentication is added if this option is selected.

InternalNetwork

ExternalNetwork

EncryptedTCP Session

(a) Transport-level security

Internet

CorporateNetwork Corporate

Network

CorporateNetwork

CorporateNetwork

(b) A virtual private network via tunnel mode

Encrypted tunnelscarrying IP traffic

Figure 20.7 Transport-Mode versus Tunnel-Mode Encryption


2. The packet is then routed to the destination. Each intermediate router needs to examine and process the IP header plus any plaintext IP extension headers but does not need to examine the ciphertext.

3. The destination node examines and processes the IP header plus any plaintext IP extension headers. Then, on the basis of the SPI in the ESP header, the destination node decrypts the remainder of the packet to recover the plaintext transport-layer segment.

Orig IPhdr

Hop-by-hop, dest,routing, fragmentIPv6

Orig IPhdrIPv4

New IPhdrIPv4

(b) Transport Mode

New IPhdr

ExtheadersIPv6

authenticated

encrypted

authenticated

encrypted

authenticated

encrypted

authenticated

encrypted

(c) Tunnel Mode

Orig IPhdr

Extheaders TCP Data

ESPtrlr

ESPauth

ESPhdr

ESPauth

Orig IPhdr TCP Data

ESPtrlr

ESPauth

ESPhdr

Dest TCP Data

TCP Data

ESPtrlr

ESPauth

ESPtrlr

ESPhdr

ESPhdr

Orig IPhdr

Extension headers(if present) TCP DataIPv6

Orig IPhdr TCP DataIPv4

(a) Before Applying ESP

Figure 20.8 Scope of ESP Encryption and Authentication


Transport mode operation provides confidentiality for any application that uses it, thus avoiding the need to implement confidentiality in every individual ap-plication. One drawback to this mode is that it is possible to do traffic analysis on the transmitted packets.

Tunnel Mode esp Tunnel mode ESP is used to encrypt an entire IP packet (Figure 20.8c). For this mode, the ESP header is prefixed to the packet and then the packet plus the ESP trailer is encrypted. This method can be used to counter traffic analysis.

Because the IP header contains the destination address and possibly source routing directives and hop-by-hop option information, it is not possible simply to transmit the encrypted IP packet prefixed by the ESP header. Intermediate routers would be unable to process such a packet. Therefore, it is necessary to encapsulate the entire block (ESP header plus ciphertext plus Authentication Data, if present) with a new IP header that will contain sufficient information for routing but not for traffic analysis.

Whereas the transport mode is suitable for protecting connections between hosts that support the ESP feature, the tunnel mode is useful in a configuration that includes a firewall or other sort of security gateway that protects a trusted network from external networks. In this latter case, encryption occurs only between an ex-ternal host and the security gateway or between two security gateways. This relieves hosts on the internal network of the processing burden of encryption and simplifies the key distribution task by reducing the number of needed keys. Further, it thwarts traffic analysis based on ultimate destination.

Consider a case in which an external host wishes to communicate with a host on an internal network protected by a firewall, and in which ESP is implemented in the external host and the firewalls. The following steps occur for transfer of a transport-layer segment from the external host to the internal host.

1. The source prepares an inner IP packet with a destination address of the tar-get internal host. This packet is prefixed by an ESP header; then the packet and ESP trailer are encrypted and Authentication Data may be added. The re-sulting block is encapsulated with a new IP header (base header plus optional extensions such as routing and hop-by-hop options for IPv6) whose destina-tion address is the firewall; this forms the outer IP packet.

2. The outer packet is routed to the destination firewall. Each intermediate router needs to examine and process the outer IP header plus any outer IP extension headers but does not need to examine the ciphertext.

3. The destination firewall examines and processes the outer IP header plus any outer IP extension headers. Then, on the basis of the SPI in the ESP header, the destination node decrypts the remainder of the packet to recover the plaintext inner IP packet. This packet is then transmitted in the internal network.

4. The inner packet is routed through zero or more routers in the internal net-work to the destination host.

Figure 20.9 shows the protocol architecture for the two modes.

20.4 / COmbInIng SeCurIty aSSOCIatIOnS 645

20.4 cOmbining security assOciatiOns

An individual SA can implement either the AH or ESP protocol but not both. Sometimes a particular traffic flow will call for the services provided by both AH and ESP. Further, a particular traffic flow may require IPsec services between hosts and, for that same flow, separate services between security gateways, such as

Data

Data

TCPhdr

Data

TCPhdr

DataOrig IP

hdr

TCPhdr

DataESPtrlr

ESPhdr

Orig IPhdr

ESPauth

New IPhdr

TCPhdr Data

ESPtrlr

ESPhdr

Orig IPhdr

ESPauth

TCPhdr

Data

Orig IPhdr

TCPhdr

Data

Orig IPhdr

TCPhdr

Data

(a) Transport mode

(b) Tunnel mode

ESPtrlr

ESPhdr

ESPauth

Application

TCP

IP

IPsec

Application

TCP

IP

IPsec

IP

Figure 20.9 Protocol Operation for ESP


firewalls. In all of these cases, multiple SAs must be employed for the same traffic flow to achieve the desired IPsec services. The term security association bundle re-fers to a sequence of SAs through which traffic must be processed to provide a de-sired set of IPsec services. The SAs in a bundle may terminate at different endpoints or at the same endpoints.

Security associations may be combined into bundles in two ways:

• Transport adjacency: Refers to applying more than one security protocol to the same IP packet without invoking tunneling. This approach to combining AH and ESP allows for only one level of combination; further nesting yields no added benefit since the processing is performed at one IPsec instance: the (ultimate) destination.

• Iterated tunneling: Refers to the application of multiple layers of security pro-tocols effected through IP tunneling. This approach allows for multiple levels of nesting, since each tunnel can originate or terminate at a different IPsec site along the path.

The two approaches can be combined, for example, by having a transport SA be-tween hosts travel part of the way through a tunnel SA between security gateways.

One interesting issue that arises when considering SA bundles is the order in which authentication and encryption may be applied between a given pair of end-points and the ways of doing so. We examine that issue next. Then we look at com-binations of SAs that involve at least one tunnel.

Authentication Plus Confidentiality

Encryption and authentication can be combined in order to transmit an IP packet that has both confidentiality and authentication between hosts. We look at several approaches.

esp wITh auThenTIcaTIon opTIon This approach is illustrated in Figure 20.8. In this approach, the user first applies ESP to the data to be protected and then appends the authentication data field. There are actually two subcases:

• Transport mode ESP: Authentication and encryption apply to the IP payload delivered to the host, but the IP header is not protected.

• Tunnel mode ESP: Authentication applies to the entire IP packet delivered to the outer IP destination address (e.g., a firewall), and authentication is per-formed at that destination. The entire inner IP packet is protected by the pri-vacy mechanism for delivery to the inner IP destination.

For both cases, authentication applies to the ciphertext rather than the plaintext.

TransporT adjacency Another way to apply authentication after encryption is to use two bundled transport SAs, with the inner being an ESP SA and the outer being an AH SA. In this case, ESP is used without its authentication option. Because the inner SA is a transport SA, encryption is applied to the IP payload. The resulting packet consists of an IP header (and possibly IPv6 header extensions) followed by an ESP. AH is then applied in transport mode, so that authentication covers

20.4 / COmbInIng SeCurIty aSSOCIatIOnS 647

the ESP plus the original IP header (and extensions) except for mutable fields. The advantage of this approach over simply using a single ESP SA with the ESP authentication option is that the authentication covers more fields, including the source and destination IP addresses. The disadvantage is the overhead of two SAs versus one SA.

TransporT-Tunnel bundle The use of authentication prior to encryption might be preferable for several reasons. First, because the authentication data are pro-tected by encryption, it is impossible for anyone to intercept the message and alter the authentication data without detection. Second, it may be desirable to store the authentication information with the message at the destination for later reference. It is more convenient to do this if the authentication information applies to the un-encrypted message; otherwise the message would have to be reencrypted to verify the authentication information.

One approach to applying authentication before encryption between two hosts is to use a bundle consisting of an inner AH transport SA and an outer ESP tunnel SA. In this case, authentication is applied to the IP payload plus the IP header (and extensions) except for mutable fields. The resulting IP packet is then processed in tunnel mode by ESP; the result is that the entire, authenticated inner packet is en-crypted and a new outer IP header (and extensions) is added.

Basic Combinations of Security Associations

The IPsec Architecture document lists four examples of combinations of SAs that must be supported by compliant IPsec hosts (e.g., workstation, server) or security gateways (e.g., firewall, router). These are illustrated in Figure 20.10. The lower part of each case in the figure represents the physical connectivity of the elements; the upper part represents logical connectivity via one or more nested SAs. Each SA can be either AH or ESP. For host-to-host SAs, the mode may be either transport or tunnel; otherwise it must be tunnel mode.

Case 1. All security is provided between end systems that implement IPsec. For any two end systems to communicate via an SA, they must share the appropri-ate secret keys. Among the possible combinations are

a. AH in transport mode

b. ESP in transport mode

c. ESP followed by AH in transport mode (an ESP SA inside an AH SA)

d. Any one of a, b, or c inside an AH or ESP in tunnel mode

We have already discussed how these various combinations can be used to support authentication, encryption, authentication before encryption, and authenti-cation after encryption.

Case 2. Security is provided only between gateways (routers, firewalls, etc.) and no hosts implement IPsec. This case illustrates simple virtual private network support. The security architecture document specifies that only a single tunnel SA is needed for this case. The tunnel could support AH, ESP, or ESP with the authenti-cation option. Nested tunnels are not required, because the IPsec services apply to the entire inner packet.

648

Internet

Tunnel SAOne or Two SAs

LocalIntranet

LocalIntranet

Host* Host*

SecurityGateway*

SecurityGateway*

(c) Case 3

Internet

Tunnel SA

LocalIntranet

LocalIntranet

HostHost

SecurityGateway*

SecurityGateway*

(b) Case 2

* = implements IPsec

Internet

One or More SAs

LocalIntranet

LocalIntranet

Host* Host*

Router Router

(a) Case 1

Internet LocalIntranet

Host*Host*

SecurityGateway*

(d) Case 4

Tunnel SAOne or Two SAs

Figure 20.10 Basic Combinations of Security Associations

20.5 / Internet Key exChange 649

Case 3. This builds on case 2 by adding end-to-end security. The same combi-nations discussed for cases 1 and 2 are allowed here. The gateway-to-gateway tun-nel provides either authentication, confidentiality, or both for all traffic between end systems. When the gateway-to-gateway tunnel is ESP, it also provides a lim-ited form of traffic confidentiality. Individual hosts can implement any additional IPsec services required for given applications or given users by means of end-to-end SAs.

Case 4. This provides support for a remote host that uses the Internet to reach an organization’s firewall and then to gain access to some server or workstation behind the firewall. Only tunnel mode is required between the remote host and the firewall. As in case 1, one or two SAs may be used between the remote host and the local host.

20.5 internet Key exchange

The key management portion of IPsec involves the determination and distri-bution of secret keys. A typical requirement is four keys for communication between two applications: transmit and receive pairs for both integrity and con-fidentiality. The IPsec Architecture document mandates support for two types of key management:

• Manual: A system administrator manually configures each system with its own keys and with the keys of other communicating systems. This is practical for small, relatively static environments.

• Automated: An automated system enables the on-demand creation of keys for SAs and facilitates the use of keys in a large distributed system with an evolving configuration.

The default automated key management protocol for IPsec is referred to as ISAKMP/Oakley and consists of the following elements:

• Oakley Key Determination Protocol: Oakley is a key exchange protocol based on the Diffie-Hellman algorithm but providing added security. Oakley is generic in that it does not dictate specific formats.

• Internet Security Association and Key Management Protocol (ISAKMP): ISAKMP provides a framework for Internet key management and provides the specific protocol support, including formats, for negotiation of security attributes.

ISAKMP by itself does not dictate a specific key exchange algorithm; rather, ISAKMP consists of a set of message types that enable the use of a variety of key exchange algorithms. Oakley is the specific key exchange algorithm mandated for use with the initial version of ISAKMP.

In IKEv2, the terms Oakley and ISAKMP are no longer used, and there are significant differences from the use of Oakley and ISAKMP in IKEv1. Nevertheless, the basic functionality is the same. In this section, we describe the IKEv2 specification.


Key Determination Protocol

IKE key determination is a refinement of the Diffie-Hellman key exchange algo-rithm. Recall that Diffie-Hellman involves the following interaction between users A and B. There is prior agreement on two global parameters: q, a large prime num-ber; and a, a primitive root of q. A selects a random integer XA as its private key and transmits to B its public key ΥA = aXA mod q. Similarly, B selects a random integer XB as its private key and transmits to A its public key ΥB = aXB mod q. Each side can now compute the secret session key:

K = (ΥB)XA mod q = (ΥA)XB mod q = aXAXB mod q

The Diffie-Hellman algorithm has two attractive features:

• Secret keys are created only when needed. There is no need to store secret keys for a long period of time, exposing them to increased vulnerability.

• The exchange requires no pre-existing infrastructure other than an agreement on the global parameters.

However, there are a number of weaknesses to Diffie-Hellman, as pointed out in [HUIT98].

• It does not provide any information about the identities of the parties.

• It is subject to a man-in-the-middle attack, in which a third party C imperson-ates B while communicating with A and impersonates A while communicating with B. Both A and B end up negotiating a key with C, which can then listen to and pass on traffic. The man-in-the-middle attack proceeds as

1. B sends his public key YB in a message addressed to A (see Figure 10.2).

2. The enemy (E) intercepts this message. E saves B’s public key and sends a message to A that has B’s User ID but E’s public key YE. This message is sent in such a way that it appears as though it was sent from B’s host system. A receives E’s message and stores E’s public key with B’s User ID. Similarly, E sends a message to B with E’s public key, purporting to come from A.

3. B computes a secret key K1 based on B’s private key and YE. A computes a secret key K2 based on A’s private key and YE. E computes K1 using E’s secret key XE and YB and computers K2 using XE and YA.

4. From now on, E is able to relay messages from A to B and from B to A, ap-propriately changing their encipherment en route in such a way that neither A nor B will know that they share their communication with E.

• It is computationally intensive. As a result, it is vulnerable to a clogging attack, in which an opponent requests a high number of keys. The victim spends con-siderable computing resources doing useless modular exponentiation rather than real work.

IKE key determination is designed to retain the advantages of Diffie-Hellman, while countering its weaknesses.


FeaTures oF Ike key deTerMInaTIon The IKE key determination algorithm is characterized by five important features:

1. It employs a mechanism known as cookies to thwart clogging attacks.

2. It enables the two parties to negotiate a group; this, in essence, specifies the global parameters of the Diffie-Hellman key exchange.

3. It uses nonces to ensure against replay attacks.

4. It enables the exchange of Diffie-Hellman public key values.

5. It authenticates the Diffie-Hellman exchange to thwart man-in-the-middle attacks.

We have already discussed Diffie-Hellman. Let us look at the remainder of these elements in turn. First, consider the problem of clogging attacks. In this attack, an opponent forges the source address of a legitimate user and sends a public Diffie-Hellman key to the victim. The victim then performs a modu-lar exponentiation to compute the secret key. Repeated messages of this type can clog the victim’s system with useless work. The cookie exchange requires that each side send a pseudorandom number, the cookie, in the initial message, which the other side acknowledges. This acknowledgment must be repeated in the first message of the Diffie-Hellman key exchange. If the source address was forged, the opponent gets no answer. Thus, an opponent can only force a user to generate acknowledgments and not to perform the Diffie-Hellman calculation.

IKE mandates that cookie generation satisfy three basic requirements:

1. The cookie must depend on the specific parties. This prevents an attacker from obtaining a cookie using a real IP address and UDP port and then using it to swamp the victim with requests from randomly chosen IP addresses or ports.

2. It must not be possible for anyone other than the issuing entity to generate cookies that will be accepted by that entity. This implies that the issuing entity will use local secret information in the generation and subsequent verification of a cookie. It must not be possible to deduce this secret information from any particular cookie. The point of this requirement is that the issuing entity need not save copies of its cookies, which are then more vulnerable to discovery, but can verify an incoming cookie acknowledgment when it needs to.

3. The cookie generation and verification methods must be fast to thwart attacks intended to sabotage processor resources.

The recommended method for creating the cookie is to perform a fast hash (e.g., MD5) over the IP Source and Destination addresses, the UDP Source and Destination ports, and a locally generated secret value.

IKE key determination supports the use of different groups for the Diffie-Hellman key exchange. Each group includes the definition of the two global pa-rameters and the identity of the algorithm. The current specification includes the following groups.


• Modular exponentiation with a 768-bit modulus

q = 2768 - 2704 - 1 + 264 * (:2638 * p; + 149686)

a = 2


q = 21024 - 2960 - 1 + 264 * (:2894 * p; + 129093)

a = 2


• Parameters to be determined

• Elliptic curve group over 2155

• Generator (hexadecimal): X = 7B, Y = 1C8

• Elliptic curve parameters (hexadecimal): A = 0, Y = 7338F

• Elliptic curve group over 2185

• Generator (hexadecimal): X = 18, Y = D

• Elliptic curve parameters (hexadecimal): A = 0, Y = 1EE9

The first three groups are the classic Diffie-Hellman algorithm using modular exponentiation. The last two groups use the elliptic curve analog to Diffie-Hellman, which was described in Chapter 10.

IKE key determination employs nonces to ensure against replay attacks. Each nonce is a locally generated pseudorandom number. Nonces appear in responses and are encrypted during certain portions of the exchange to secure their use.

Three different authentication methods can be used with IKE key determination:

• Digital signatures: The exchange is authenticated by signing a mutually ob-tainable hash; each party encrypts the hash with its private key. The hash is generated over important parameters, such as user IDs and nonces.

• Public-key encryption: The exchange is authenticated by encrypting param-eters such as IDs and nonces with the sender’s private key.

• Symmetric-key encryption: A key derived by some out-of-band mechanism can be used to authenticate the exchange by symmetric encryption of ex-change parameters.

Ikev2 exchanges The IKEv2 protocol involves the exchange of messages in pairs. The first two pairs of exchanges are referred to as the initial exchanges (Figure 20.11a). In the first exchange, the two peers exchange information concern-ing cryptographic algorithms and other security parameters they are willing to use along with nonces and Diffie-Hellman (DH) values. The result of this exchange is to set up a special SA called the IKE SA (see Figure 20.2). This SA defines parameters for a secure channel between the peers over which subsequent message exchanges take place. Thus, all subsequent IKE message exchanges are protected by encryp-tion and message authentication. In the second exchange, the two parties authenti-cate one another and set up a first IPsec SA to be placed in the SADB and used for


protecting ordinary (i.e. non-IKE) communications between the peers. Thus, four messages are needed to establish the first SA for general use.

The CREATE_CHILD_SA exchange can be used to establish further SAs for protecting traffic. The informational exchange is used to exchange management information, IKEv2 error messages, and other notifications.

Header and Payload Formats

IKE defines procedures and packet formats to establish, negotiate, modify, and de-lete security associations. As part of SA establishment, IKE defines payloads for exchanging key generation and authentication data. These payload formats provide a consistent framework independent of the specific key exchange protocol, encryp-tion algorithm, and authentication mechanism.

Ike header ForMaT An IKE message consists of an IKE header followed by one or more payloads. All of this is carried in a transport protocol. The specification dic-tates that implementations must support the use of UDP for the transport protocol.

HDR, SAi1, KEi, Ni

rednopseRrotaitinI

(a) Initial exchanges

HDR, SAr1, KEr, Nr, [CERTREQ]

HDR, SK {IDi, [CERT,] [CERTREQ,] [IDr,] AUTH, SAi2, TSi, TSr}

HDR, SK {IDr, [CERT,] AUTH, SAr2, TSi, TSr}

HDR, SK {[N], SA, Ni, [KEi], [TSi, TSr]}

(b) CREATE_CHILD_SA exchange

HDR, SK {SA, Nr, [KEr], [TSi, TSr]}

HDR, SK {[N,] [D,] [CP,] ...}

(c) Informational exchange

HDR, SK {[N,] [D,] [CP], ...}

HDR = IKE headerSAx1 = offered and chosen algorithms, DH groupKEx = Diffie-Hellman public keyNx= noncesCERTREQ = Certificate requestIDx = identityCERT = certificate

SK {...} = MAC and encryptAUTH = AuthenticationSAx2 = algorithms, parameters for IPsec SATSx = traffic selectors for IPsec SAN = NotifyD = DeleteCP = Configuration

Figure 20.11 IKEv2 Exchanges


Figure 20.12a shows the header format for an IKE message. It consists of the following fields.

• Initiator SPI (64 bits): A value chosen by the initiator to identify a unique IKE security association (SA).

• Responder SPI (64 bits): A value chosen by the responder to identify a unique IKE SA.

• Next Payload (8 bits): Indicates the type of the first payload in the message; payloads are discussed in the next subsection.

• Major Version (4 bits): Indicates major version of IKE in use.

• Minor Version (4 bits): Indicates minor version in use.

• Exchange Type (8 bits): Indicates the type of exchange; these are discussed later in this section.

• Flags (8 bits): Indicates specific options set for this IKE exchange. Three bits are defined so far. The initiator bit indicates whether this packet is sent by the SA initiator. The version bit indicates whether the transmitter is capable of using a higher major version number than the one currently indicated. The response bit indicates whether this is a response to a message containing the same message ID.

• Message ID (32 bits): Used to control retransmission of lost packets and matching of requests and responses.

• Length (32 bits): Length of total message (header plus all payloads) in octets.

Ike payload Types All IKE payloads begin with the same generic payload header shown in Figure 20.12b. The Next Payload field has a value of 0 if this is the last

MjVer MnVer Exchange Type FlagsNext Payload

Message ID

Length

(a) IKE header

(b) Generic Payload header

Initiator’s Security Parameter Index (SPI)

Responder’s Security Parameter Index (SPI)

0Bit: 8 16 24 31

RESERVED Payload LengthNext Payload C

0Bit: 8 16 31

Figure 20.12 IKE Formats


payload in the message; otherwise its value is the type of the next payload. The Payload Length field indicates the length in octets of this payload, including the generic payload header.

The critical bit is 0 if the sender wants the recipient to skip this payload if it does not understand the payload type code in the Next Payload field of the previous payload. It is set to 1 if the sender wants the recipient to reject this entire message if it does not understand the payload type.

Table 20.3 summarizes the payload types defined for IKE and lists the fields, or parameters, that are part of each payload. The SA payload is used to begin the establishment of an SA. The payload has a complex, hierarchical structure. The payload may contain multiple proposals. Each proposal may contain multiple pro-tocols. Each protocol may contain multiple transforms. And each transform may contain multiple attributes. These elements are formatted as substructures within the payload as follows.

• Proposal: This substructure includes a proposal number, a protocol ID (AH, ESP, or IKE), an indicator of the number of transforms, and then a trans-form substructure. If more than one protocol is to be included in a proposal, then there is a subsequent proposal substructure with the same proposal number.

• Transform: Different protocols support different transform types. The trans-forms are used primarily to define cryptographic algorithms to be used with a particular protocol.

• Attribute: Each transform may include attributes that modify or complete the specification of the transform. An example is key length.

Table 20.3 IKE Payload Types

Type Parameters

Security Association Proposals

Key Exchange DH Group #, Key Exchange Data

Identification ID Type, ID Data

Certificate Cert Encoding, Certificate Data

Certificate Request Cert Encoding, Certification Authority

Authentication Auth Method, Authentication Data

Nonce Nonce Data

Notify Protocol-ID, SPI Size, Notify Message Type, SPI, Notification Data

Delete Protocol-ID, SPI Size, # of SPIs, SPI (one or more)

Vendor ID Vendor ID

Traffic Selector Number of TSs, Traffic Selectors

Encrypted IV, Encrypted IKE payloads, Padding, Pad Length, ICV

Configuration CFG Type, Configuration Attributes

Extensible Authentication Protocol

EAP Message


The Key Exchange payload can be used for a variety of key exchange tech-niques, including Oakley, Diffie-Hellman, and the RSA-based key exchange used by PGP. The Key Exchange data field contains the data required to generate a ses-sion key and is dependent on the key exchange algorithm used.

The Identification payload is used to determine the identity of communicating peers and may be used for determining authenticity of information. Typically the ID Data field will contain an IPv4 or IPv6 address.

The Certificate payload transfers a public-key certificate. The Certificate Encoding field indicates the type of certificate or certificate-related information, which may include the following:

• PKCS #7 wrapped X.509 certificate

• PGP certificate

• DNS signed key

• X.509 certificate—signature

• X.509 certificate—key exchange

• Kerberos tokens

• Certificate Revocation List (CRL)

• Authority Revocation List (ARL)

• SPKI certificate

At any point in an IKE exchange, the sender may include a Certificate Request payload to request the certificate of the other communicating entity. The payload may list more than one certificate type that is acceptable and more than one certifi-cate authority that is acceptable.

The Authentication payload contains data used for message authentication purposes. The authentication method types so far defined are RSA digital signa-ture, shared-key message integrity code, and DSS digital signature.

The Nonce payload contains random data used to guarantee liveness during an exchange and to protect against replay attacks.

The Notify payload contains either error or status information associated with this SA or this SA negotiation. The following table lists the IKE notify messages.

Error Messages Status Messages

Unsupported Critical Initial Contact

Payload Set Window Size

Invalid IKE SPI Additional TS Possible

Invalid Major Version IPCOMP Supported

Invalid Syntax NAT Detection Source IP

Invalid Payload Type NAT Detection Destination IP

Invalid Message ID Cookie

Invalid SPI Use Transport Mode

20.6 / CryptOgraphIC SuIteS 657

Error Messages Status Messages

No Proposal Chosen HTTP Cert Lookup Supported

Invalid KE Payload Rekey SA

Authentication Failed ESP TFC Padding Not Supported

Single Pair Required Non First Fragments Also

No Additional SAS

Internal Address Failure

Failed CP Required

TS Unacceptable

Invalid Selectors

The Delete payload indicates one or more SAs that the sender has deleted from its database and that therefore are no longer valid.

The Vendor ID payload contains a vendor-defined constant. The constant is used by vendors to identify and recognize remote instances of their implementa-tions. This mechanism allows a vendor to experiment with new features while main-taining backward compatibility.

The Traffic Selector payload allows peers to identify packet flows for process-ing by IPsec services.

The Encrypted payload contains other payloads in encrypted form. The en-crypted payload format is similar to that of ESP. It may include an IV if the encryp-tion algorithm requires it and an ICV if authentication is selected.

The Configuration payload is used to exchange configuration information be-tween IKE peers.

The Extensible Authentication Protocol (EAP) payload allows IKE SAs to be authenticated using EAP, which was discussed in Chapter 16.

20.6 cryPtOgraPhic suites

The IPsecv3 and IKEv3 protocols rely on a variety of types of cryptographic algo-rithms. As we have seen in this book, there are many cryptographic algorithms of each type, each with a variety of parameters, such as key size. To promote interop-erability, two RFCs define recommended suites of cryptographic algorithms and parameters for various applications.

RFC 4308 defines two cryptographic suites for establishing virtual private net-works. Suite VPN-A matches the commonly used corporate VPN security used in older IKEv1 implementations at the time of the issuance of IKEv2 in 2005. Suite VPN-B provides stronger security and is recommended for new VPNs that imple-ment IPsecv3 and IKEv2.

Table 20.4a lists the algorithms and parameters for the two suites. There are several points to note about these two suites. Note that for symmetric cryptography,


VPN-A relies on 3DES and HMAC, while VPN-B relies exclusively on AES. Three types of secret-key algorithms are used:

• Encryption: For encryption, the cipher block chaining (CBC) mode is used.

• Message authentication: For message authentication, VPN-A relies on HMAC with SHA-1 with the output truncated to 96 bits. VPN-B relies on a variant of CMAC with the output truncated to 96 bits.

• Pseudorandom function: IKEv2 generates pseudorandom bits by repeated use of the MAC used for message authentication.

RFC 6379 defines four optional cryptographic suites that are compatible with the United States National Security Agency’s Suite B specifications. In 2005, the NSA issued Suite B, which defined the algorithms and strengths needed to pro-tect both sensitive but unclassified (SBU) and classified information for use in its Cryptographic Modernization program [LATT09]. The four suites defined in RFC 4869 provide choices for ESP and IKE. The four suites are differentiated by the choice of cryptographic algorithm strengths and a choice of whether ESP is to provide both confidentiality and integrity or integrity only. All of the suites offer greater protection than the two VPN suites defined in RFC 4308.

Table 20.4 Cryptographic Suites for IPsec

VPN-A VPN-B

ESP encryption 3DES-CBC AES-CBC (128-bit key)

ESP integrity HMAC-SHA1-96 AES-XCBC-MAC-96

IKE encryption 3DES-CBC AES-CBC (128-bit key)

IKE PRF HMAC-SHA1 AES-XCBC-PRF-128

IKE Integrity HMAC-SHA1-96 AES-XCBC-MAC-96

IKE DH group 1024-bit MODP 2048-bit MODP

(a) Virtual private networks (RFC 4308)

GCM-128 GCM-256 GMAC-128 GMAC-256

ESP encryption/ Integrity

AES-GCM (128-bit key)

AES-GCM (256-bit key)

Null Null

ESP integrity Null Null AES-GMAC (128-bit key)

AES-GMAC (256-bit key)

IKE encryption AES-CBC (128-bit key)

AES-CBC (256-bit key)



IKE PRF HMAC-SHA-256 HMAC-SHA-384 HMAC-SHA-256 HMAC-SHA-384

IKE Integrity HMAC-SHA- 256-128

HMAC-SHA- 384-192

HMAC-SHA- 256-128

HMAC-SHA- 384-192

IKE DH group 256-bit random ECP

384-bit random ECP

256-bit random ECP

384-bit random ECP

(b) NSA Suite B (RFC 4869)

20.8 / Key termS, revIew QueStIOnS, and prOblemS 659

Table 20.4b lists the algorithms and parameters for the two suites. As with RFC 4308, three categories of secret key algorithms are listed:

• Encryption: For ESP, authenticated encryption is provided using the GCM mode with either 128-bit or 256-bit AES keys. For IKE encryption, CBC is used, as it was for the VPN suites.

• Message authentication: For ESP, if only authentication is required, then GMAC is used. As discussed in Chapter 12, GMAC is simply the authenti-cation portion of GMC. For IKE, message authentication is provided using HMAC with one of the SHA-3 hash functions.

• Pseudorandom function: As with the VPN suites, IKEv2 in these suites gen-erates pseudorandom bits by repeated use of the MAC used for message authentication.

For the Diffie-Hellman algorithm, the use of elliptic curve groups modulo a prime is specified. For authentication, elliptic curve digital signatures are listed. The original IKEv2 documents used RSA-based digital signatures. Equivalent or greater strength can be achieved using ECC with fewer key bits.


IPv6 and IPv4 are covered in more detail in [STAL11]. [CHEN98] provides a good discussion of an IPsec design. [FRAN05] is a more comprehensive treatment of IPsec. [PATE06] is a useful overview of IPsecv3 and IKEv2 with an emphasis on cryptographic aspects.

CHEN98 Cheng, P., et al. “A Security Architecture for the Internet Protocol.” IBM Systems Journal, Number 1, 1998.

FRAN05 Frankel, S., et al. Guide to IPsec VPNs. NIST SP 800-77, 2005.PATE06 Paterson, K. “A Cryptographic Tour of the IPsec Standards.” Cryptology ePrint

Archive: Report 2006/097, April 2006.STAL11 Stallings, W. Data and Computer Communications, Ninth Edition. Upper

Saddle River, NJ: Prentice Hall, 2011.


Key Terms

anti-replay serviceAuthentication Header (AH)Encapsulating Security

Payload (ESP)Internet Security Association

and Key Management Protocol (ISAKMP)

Internet Key Exchange (IKE)IP Security (IPsec)IPv4IPv6Oakley key determination

protocolreplay attack

security association (SA)transport modetunnel mode


Review Questions 20.1 Give examples of applications of IPsec. 20.2 What services are provided by IPsec? 20.3 What parameters identify an SA and what parameters characterize the nature of a

particular SA? 20.4 What is the difference between transport mode and tunnel mode? 20.5 What is a replay attack? 20.6 Why does ESP include a padding field? 20.7 What are the basic approaches to bundling SAs? 20.8 What are the roles of the Oakley key determination protocol and ISAKMP in IPsec?

Problems 20.1 Describe and explain each of the entries in Table 20.2. 20.2 Draw a figure similar to Figure 20.8 for AH. 20.3 List the major security services provided by AH and ESP, respectively. 20.4 In discussing AH processing, it was mentioned that not all of the fields in an IP header

are included in MAC calculation. a. For each of the fields in the IPv4 header, indicate whether the field is immutable,

mutable but predictable, or mutable (zeroed prior to ICV calculation). b. Do the same for the IPv6 header. c. Do the same for the IPv6 extension headers.

In each case, justify your decision for each field. 20.5 Suppose that the current replay window spans from 120 to 530. a. If the next incoming authenticated packet has sequence number 105, what will the re-

ceiver do with the packet, and what will be the parameters of the window after that? b. If instead the next incoming authenticated packet has sequence number 440, what

will the receiver do with the packet, and what will be the parameters of the win-dow after that?

c. If instead the next incoming authenticated packet has sequence number 540, what will the receiver do with the packet, and what will be the parameters of the win-dow after that?

20.6 When tunnel mode is used, a new outer IP header is constructed. For both IPv4 and IPv6, indicate the relationship of each outer IP header field and each extension header in the outer packet to the corresponding field or extension header of the inner IP packet. That is, indicate which outer values are derived from inner values and which are constructed independently of the inner values.

20.7 End-to-end authentication and encryption are desired between two hosts. Draw figures similar to Figure 20.8 that show each of the following.

a. Transport adjacency with encryption applied before authentication. b. A transport SA bundled inside a tunnel SA with encryption applied before

authentication. c. A transport SA bundled inside a tunnel SA with authentication applied before

encryption. 20.8 The IPsec architecture document states that when two transport mode SAs are bun-

dled to allow both AH and ESP protocols on the same end-to-end flow, only one ordering of security protocols seems appropriate: performing the ESP protocol be-fore performing the AH protocol. Why is this approach recommended rather than authentication before encryption?

20.9 For the IKE key exchange, indicate which parameters in each message go in which ISAKMP payload types.

20.10 Where does IPsec reside in a protocol stack?

661

Appendix A

projects for teAching cryptogrAphy And network security

A.1 Sage Computer Algebra Projects

A.2 Hacking Project

A.3 Block Cipher Projects

A.4 Laboratory Exercises

A.5 Research Projects

A.6 Programming Projects

A.7 Practical Security Assessments

A.8 Firewall Projects

A.9 Case Studies

A.10 Writing Assignments

A.11 Reading/Report Assignments

A.12 Discussion Topics

661

662 Appendix A / projects for teAching cryptogrAphy

Analysis and observation, theory and experience must never disdain or exclude each other; on the contrary, they support each other.

—On War, Carl Von Clausewitz

Many instructors believe that research or implementation projects are crucial to the clear understanding of cryptography and network security. Without projects, it may be difficult for students to grasp some of the basic concepts and interactions among components. Projects reinforce the concepts introduced in the book, give the stu-dent a greater appreciation of how a cryptographic algorithm or protocol works, and can motivate students and give them confidence that they are capable of not only understanding but implementing the details of a security capability.

In this text, I have tried to present the concepts of cryptography and network security as clearly as possible and have provided numerous homework problems to reinforce those concepts. However, many instructors will wish to supplement this material with projects. This appendix provides some guidance in that regard and describes support material available in the Instructor’s Resource Center (IRC) for this book, accessible to instructors from Prentice Hall. The support material covers 12 types of projects and other student exercises:

• Sage computer algebra projects

• Hacking project

• Block cipher projects

• Laboratory exercises

• Research projects

• Programming projects

• Practical security assessments

• Firewall projects

• Case studies

• Writing assignments

• Reading/report assignments

• Discussion topics

A.1 SAge Computer AlgebrA projeCtS

One of the most important new features for this edition is the use of Sage for cryp-tographic examples and homework assignments. Sage is an open-source, multiplat-form, freeware package that implements a very powerful, flexible, and easily learned mathematics and computer algebra system. A computer algebra system (CAS) is software that can perform symbolic as well as numerical calculations. CASs have been used for teaching since their inception some decades ago, and there is now a considerable literature on their use. A CAS is a natural tool for extending the learn-ing experience in a cryptography course.

A.2 / hAcking project 663

Unlike competing systems such as Mathematica, Maple, and MATLAB, there are no licensing agreements or fees involved with Sage. Thus, Sage can be made avail-able on computers and networks at school, and students can individually download the software to their own personal computers for use at home. Another advantage of using Sage is that students learn a powerful, flexible tool that can be used for virtually any mathematical application, not just cryptography. The Sage Web site (http://www .sagemath.org) provides considerable documentation on how to install, set up, and use Sage on a variety of computers and how to use it online via the Web.

The use of Sage can make a significant difference to the teaching of the mathematics of cryptographic algorithms. Appendix B provides a large number of examples of the use of Sage covering many cryptographic concepts. Appendix C lists exercises in each of these topic areas to enable the student to gain hands-on experience with cryptographic algorithms. Copies of both appendices are available online so that students do not have to key in lines of code that are printed in the appendices.

The IRC contains solutions to all of the exercises in Appendix C.Dan Shumow of Microsoft and the University of Washington developed all of

the examples and assignments in Appendices B and C.

A.2 HACking projeCt

The aim of this project is to hack into a corporation’s network through a series of steps. The Corporation is named Extreme In Security Corporation. As the name indicates, the corporation has some security holes in it, and a clever hacker is able to access critical information by hacking into its network. The IRC includes what is needed to set up the Web site. The student’s goal is to capture the secret informa-tion about the price on the quote the corporation is placing next week to obtain a contract for a governmental project.

The student should start at the Web site and find his or her way into the net-work. At each step, if the student succeeds, there are indications as to how to pro-ceed on to the next step as well as the grade until that point.

The project can be attempted in three ways:

1. Without seeking any sort of help

2. Using some provided hints

3. Using exact directions

The IRC includes the files needed for this project:

1. Web Security project

2. Web Hacking exercises (XSS and Script-attacks) covering client-side and server-side vulnerability exploitations, respectively

3. Documentation for installation and use for the above

4. A PowerPoint file describing Web hacking. This file is crucial to understand-ing how to use the exercises since it clearly explains the operation using screen shots.

http://www.sagemath.org

http://www.sagemath.org


This project was designed and implemented by Professor Sreekanth Malladi of Dakota State University.

A.3 bloCk CipHer projeCtS

This is a lab that explores the operation of the AES encryption algorithm by tracing its execution, computing one round by hand, and then exploring the various block cipher modes of use. The lab also covers DES. In both cases, an online Java applet is used (or can be downloaded) to execute AES or DES.

For both AES and DES, the lab is divided into three separate parts:

• Block cipher internals: This part involves encrypting plaintext and analyz-ing the intermediate results after each round. There is an online calculator for both AES and DES that provides the intermediate results and the final ciphertext.

• Block cipher round: This part involves calculating one round by hand and comparing the results to those produced by the calculator.

• Block cipher modes of use: Enables the student to compare the operation of CBC and CFB modes.

The IRC contains the .html and .jar files needed to set up these labs on your own Web site. Alternatively, the material is available from the Student Resources section of this book’s Web site. Click on the rotating globe.

Lawrie Brown of the Australian Defence Force Academy developed these projects.

A.4 lAborAtory exerCiSeS

Professor Sanjay Rao and Ruben Torres of Purdue University have prepared a set of laboratory exercises that are included in the IRC. These are implementation projects designed to be programmed on Linux but could be adapted for any Unix environment. These laboratory exercises provide realistic experience in implement-ing security functions and applications.

A.5 reSeArCH projeCtS

An effective way of reinforcing basic concepts from the course and for teaching students research skills is to assign a research project. Such a project could involve a literature search as well as an Internet search of vendor products, research lab activities, and standardization efforts. Projects could be assigned to teams or, for smaller projects, to individuals. In any case, it is best to require some sort of project proposal early in the term, giving the instructor time to evaluate the proposal for appropriate topic and appropriate level of effort. Student handouts for research projects should include

A.7 / prActicAl security Assessments 665

• A format for the proposal

• A format for the final report

• A schedule with intermediate and final deadlines

• A list of possible project topics

The students can select one of the topics listed in the IRC or devise their own comparable project. The IRC includes a suggested format for the proposal and final report as well as a list of 15 possible research topics.

A.6 progrAmming projeCtS

The programming project is a useful pedagogical tool. There are several attractive features of stand-alone programming projects that are not part of an existing secu-rity facility:

1. The instructor can choose from a wide variety of cryptography and network security concepts to assign projects.

2. The projects are platform and language independent. Students can program the projects on any available computer and in any appropriate language.

3. The instructor need not download, install, and configure any particular infra-structure for stand-alone projects.

There is also flexibility in the size of projects. Larger projects give students more a sense of achievement, but students with less ability or fewer organizational skills can be left behind. Larger projects usually elicit more overall effort from the best students. Smaller projects can have a higher concepts-to-code ratio and, because more of them can be assigned, the opportunity exists to address a variety of different areas.

Again, as with research projects, the students should first submit a proposal. The student handout should include the same elements listed in the preceding sec-tion. The IRC includes a set of 12 possible programming projects.

The following individuals have supplied the research and programming proj-ects suggested in the IRC: Henning Schulzrinne of Columbia University; Cetin Kaya Koc of Oregon State University; and David M. Balenson of Trusted Information Systems and George Washington University.

A.7 prACtiCAl SeCurity ASSeSSmentS

Examining the current infrastructure and practices of an existing organization is one of the best ways of developing skills in assessing its security posture. The IRC contains a list of such activities. Students, working either individually or in small groups, select a suitable small- to medium-sized organization. They then interview some key personnel in that organization in order to conduct a suitable selection of security risk assessment and review tasks as it relates to the organization’s IT infrastructure and practices. As a result, they can then recommend suitable changes,


which can improve the organization’s IT security. These activities help students develop an appreciation of current security practices and the skills needed to review these and recommend changes.

Lawrie Brown of the Australian Defence Force Academy developed these projects.

A.8 FirewAll projeCtS

The implementation of network firewalls can be a difficult concept for students to grasp initially. The IRC includes a Network Firewall Visualization tool to convey and teach network security and firewall configuration. This tool is intended to teach and reinforce key concepts including the use and purpose of a perimeter firewall, the use of separated subnets, the purposes behind packet filtering, and the short-comings of a simple packet filter firewall.

The IRC includes a .jar file that is fully portable, and a series of exercises. The tool and exercises were developed at U.S. Air Force Academy.

A.9 CASe StudieS

Teaching with case studies engages students in active learning. The IRC includes case studies in the following areas:

• Disaster recovery

• Firewalls

• Incidence response

• Physical security

• Risk

• Security policy

• Virtualization

Each case study includes learning objectives, case description, and a series of case discussion questions. Each case study is based on real-world situations and includes papers or reports describing the case.

The case studies were developed at North Carolina A&T State University.

A.10 writing ASSignmentS

Writing assignments can have a powerful multiplier effect in the learning process in a technical discipline such as cryptography and network security. Adherents of the Writing Across the Curriculum (WAC) movement (http://wac.colostate.edu/) report substantial benefits of writing assignments in facilitating learning. Writing assignments lead to more detailed and complete thinking about a particular topic. In addition, writing assignments help to overcome the tendency of students to pursue

http://wac.colostate.edu/

A.12 / discussion topics 667

a subject with a minimum of personal engagement, just learning facts and problem-solving techniques without obtaining a deep understanding of the subject matter.

The IRC contains a number of suggested writing assignments, organized by chapter. Instructors may ultimately find that this is an important part of their approach to teaching the material. I would greatly appreciate any feedback on this area and any suggestions for additional writing assignments.

A.11 reAding/report ASSignmentS

Another excellent way to reinforce concepts from the course and to give students research experience is to assign papers from the literature to be read and analyzed. The student is then asked to write a brief report on the assigned paper. The IRC includes a suggested list of papers, one or two per chapter, to be assigned. The IRC provides a PDF copy of each of the papers. The IRC also includes a suggested assignment wording.

A.12 diSCuSSion topiCS

One way to provide a collaborative experience is discussion topics, a number of which are included in the IRC. Each topic relates to material in the book. The instructor can set it up so that students can discuss a topic either in a class setting, an online chat room, or a message board. Again, I would greatly appreciate any feedback on this area and any suggestions for additional discussion topics.

668668

By Dan Shumow University of Washington

B.1 Linear Algebra and Matrix Functionality

B.2 Chapter 2: Classical Encryption

B.3 Chapter 3: Block Ciphers and the Data Encryption Standard

B.4 Chapter 4: Basic Concepts in Number Theory and Finite Fields

B.5 Chapter 5: Advanced Encryption Standard

B.6 Chapter 6: Pseudorandom Number Generation and Stream Ciphers

B.7 Chapter 8: Number Theory

B.8 Chapter 9: Public-Key Cryptography and RSA

B.9 Chapter 10: Other Public-Key Cryptosystems

B.10 Chapter 11: Cryptographic Hash Functions

B.11 Chapter 13: Digital Signatures

Appendix B

SAge exAmpleS

B.1 / Linear aLgeBra and Matrix FunctionaLity 669

This appendix contains a number of examples that illustrate cryptographic concepts, organized by the chapter in which those concepts were discussed. All the examples are in Sage.1 See Appendix C for how to get started using Sage and for a brief intro-duction to Sage syntax and operations. We begin with a brief introduction to some basic Sage matrix and linear algebra operations.

You should be able to follow the examples in this section as written. However, if you have difficulty interpreting the Sage code, please refer to Section C.2 in Appendix C.

B.1 Linear aLgeBra and Matrix FunctionaLity

Sage includes linear algebra and matrix functionality. The following shows some of the basic functionality applicable to cryptography.

In Sage you specify a matrix as a list of lists of numbers, passed to the matrix function. For example, passing a list of lists of integers as follows:

sage: M = matrix([[1, 3],[7,9]]); M

[1 3]

[7 9]

Alternately, passing a list of lists of rationals as follows:

sage: M = matrix([[1/2, 2/3, 3/4],[5, 7, 8]]); M

[1/2 2/3 3/4]

[ 5 7 8]

You can specify that the input should be reduced by a modulus, using the IntegerModRing (functionality to be described later)

Sage: R = IntegerModRing(100)

sage: M = matrix(R, [[1],[102],[1003]]); M

[1]

[2]

[3]

Or that the input should be considered in a finite field (also to be described later).

sage: F = GF(2);

sage: M = matrix(F, [[1, 2, 0, 3]]); M

[1 0 0 1]

1All of the Sage code in this appendix is available at this book’s Premium Content Web site in .sage files, so that you can load and execute the programs if you wish. See Preface for access information.

670 appendix B / Sage exaMpLeS

Sage also supports multiplication, addition, and inversion of matrices as follows:

sage: M1 = matrix([[1, 2],[3,4]]);sage: M2 = matrix([[1,-1],[1, 1]]);sage: M1*M2[3 1][7 1]

sage: M1 + M2[2 1][4 5]

sage: M2^-1

[ 1/2 1/2]

[-1/2 1/2]

B.2 chapter 2: cLassicaL encryption

The following functions are useful for classical cipher examples and exercises:

en_alphabet = "abcdefghijklmnopqrstuvwxyz"

## This function returns true if and only if the character c is an# alphabetic character#def is_alphabetic_char(c):

return (c.lower() in en_alphabet)

## This function converts a single character into its numeric value#def char_to_num(c):

return en_alphabet.index(c.lower())

## This function returns the character corresponding to x mod 26# in the English alphabet#def num_to_char(x):

return en_alphabet[x % 26]

B.2 / cLaSSicaL encryption 671

Example 1: Implement Sage encryption/decryption functions that take a key (as an integer in 0, 1, 2, . . . , 25), and a string. The function should only operate on the characters ‘a’, ‘b’, . . . ‘z’ (both upper and lower case), and it should leave any other characters unchanged.

Solution:

def CaesarEncrypt(k, plaintext):

ciphertext = ""

for j in xrange(len(plaintext)):

p = plaintext[j]

if is_alphabetic_char(p):x = (k + char_to_num(p)) % 26c = num_to_char(x)

else:

c = p

ciphertext += c

return ciphertext

def CaesarDecrypt(k, ciphertext):

plaintext = ""

for j in xrange(len(ciphertext)):

c = ciphertext[j]

if is_alphabetic_char(c):x = (char_to_num(c) - k) % 26p = num_to_char(x)

else:

p = c

plaintext += p

return plaintext

Example 2: Implement a function that performs a brute force attack on a ciphertext, it should print a list of the keys and associated decryptions. It should also take an optional parameter that takes a substring and only prints out potential plaintexts that contain that decryption.

Solution:

def BruteForceAttack(ciphertext, keyword=None):

for k in xrange(26):plaintext = CaesarDecrypt(k, ciphertext)

if (None==keyword) or (keyword in plaintext):print "key", k, "decryption", plaintext

return


Example 3: Show the output of your encrypt function (Example 1) on the fol-lowing (key, plaintext) pairs:

• k = 16 plaintext = “Get me a vanilla ice cream, make it a double.”

• k = 15 plaintext = “I don’t much care for Leonard Cohen.”

• k = 16 plaintext = “I like root beer floats.”

Solution:

sage: k = 6; plaintext = 'Get me a vanilla ice cream, make it a double.'sage: CaesarEncrypt(k, plaintext)'mkz sk g bgtorrg oik ixkgs, sgqk oz g juahrk.'

sage: k = 15; plaintext = "I don’t much care for Leonard Cohen."sage: CaesarEncrypt(k, plaintext)"x sdc'i bjrw rpgt udg atdcpgs rdwtc."

sage: k = 16; plaintext = "I like root beer floats."sage: CaesarEncrypt(k, plaintext)'y byau heej ruuh vbeqji.'

Example 4: Show the output of your decrypt function (Example 1) on the fol-lowing (key, ciphertext) pairs:

• k = 12 ciphertext = ‘nduzs ftq buzq oazqe.’

• k = 3 ciphertext = “fdhvdu qhhgv wr orvh zhljkw.”

• k = 20 ciphertext = “ufgihxm uly numnys.”

Solution:

sage: k = 12; ciphertext = "nduzs ftq buzq oazqe."sage: CaesarDecrypt(k, ciphertext)'bring the pine cones.'

sage: k = 3; ciphertext = "fdhvdu qhhgv wr orvh zhljkw."sage: CaesarDecrypt(k, ciphertext)'caesar needs to lose weight.'

sage: k = 20; ciphertext = "ufgihxm uly numnys."sage: CaesarDecrypt(k, ciphertext)'almonds are tastey.'

Example 5: Show the output of your attack function (Example 4) on the following ciphertexts, if an optional keyword is specified, pass that to your attack function:

• ciphertext = ‘gryy guru gob tab gb nzoebfr puncry.’ keyword = ‘chapel’

• ciphertext = ‘wziv kyv jyfk nyve kyv tpdsrcj tirjy.’ keyword = ‘cymbal’

• ciphertext = ‘baeeq klwosjl osk s esf ozg cfwo lgg emuz.’ no keyword

B.3 / BLock cipherS and the data encryption Standard 673

Solution:sage: ciphertext = 'gryy gurz gb tb gb nzoebfr puncry.'sage: BruteForceAttack(ciphertext, 'chapel')key 13 decryption tell them to go to ambrose chapel.

sage: ciphertext = 'wziv kyv jyfk nyve kyv tpdsrcj tirjy.'sage: BruteForceAttack(ciphertext, 'cymbal')key 17 decryption fire the shot when the cymbals crash.

sage: ciphertext = 'baeeq klwosjl osk s esf ozg cfwo lgg emuz.'sage: BruteForceAttack(ciphertext)key 0 decryption baeeq klwosjl osk s esf ozg cfwo lgg emuz.key 1 decryption azddp jkvnrik nrj r dre nyf bevn kff dlty.key 2 decryption zycco ijumqhj mqi q cqd mxe adum jee cksx.key 3 decryption yxbbn hitlpgi lph p bpc lwd zctl idd bjrw.key 4 decryption xwaam ghskofh kog o aob kvc ybsk hcc aiqv.key 5 decryption wvzzl fgrjneg jnf n zna jub xarj gbb zhpu.key 6 decryption vuyyk efqimdf ime m ymz ita wzqi faa ygot.key 7 decryption utxxj dephlce hld l xly hsz vyph ezz xfns.key 8 decryption tswwi cdogkbd gkc k wkx gry uxog dyy wemr.key 9 decryption srvvh bcnfjac fjb j vjw fqx twnf cxx vdlq.key 10 decryption rquug abmeizb eia i uiv epw svme bww uckp.key 11 decryption qpttf zaldhya dhz h thu dov ruld avv tbjo.key 12 decryption posse yzkcgxz cgy g sgt cnu qtkc zuu sain.key 13 decryption onrrd xyjbfwy bfx f rfs bmt psjb ytt rzhm.key 14 decryption nmqqc wxiaevx aew e qer als oria xss qygl.key 15 decryption mlppb vwhzduw zdv d pdq zkr nqhz wrr pxfk.key 16 decryption lkooa uvgyctv ycu c ocp yjq mpgy vqq owej.key 17 decryption kjnnz tufxbsu xbt b nbo xip lofx upp nvdi.key 18 decryption jimmy stewart was a man who knew too much.key 19 decryption ihllx rsdvzqs vzr z lzm vgn jmdv snn ltbg.key 20 decryption hgkkw qrcuypr uyq y kyl ufm ilcu rmm ksaf.key 21 decryption gfjjv pqbtxoq txp x jxk tel hkbt qll jrze.key 22 decryption feiiu opaswnp swo w iwj sdk gjas pkk iqyd.key 23 decryption edhht nozrvmo rvn v hvi rcj fizr ojj hpxc.key 24 decryption dcggs mnyquln qum u guh qbi ehyq nii gowb.key 25 decryption cbffr lmxptkm ptl t ftg pah dgxp mhh fnva.

B.3 chapter 3: BLock ciphers and the data encryption standard

Example 1: This example implements simplified DES, which is described in Appendix G.

## The Expansions/Permutations are stored as lists of bit positions#


P10_data = [3, 5, 2, 7, 4, 10, 1, 9, 8, 6];

P8_data = [6, 3, 7, 4, 8, 5, 10, 9];

LS1_data = [2, 3, 4, 5, 1];

LS2_data = [3, 4, 5, 1, 2];

IP_data = [2, 6, 3, 1, 4, 8, 5, 7];

IPinv_data = [4, 1, 3, 5, 7, 2, 8, 6];

EP_data = [4, 1, 2, 3, 2, 3, 4, 1];

P4_data = [2, 4, 3, 1];

SW_data = [5, 6, 7, 8, 1, 2, 3, 4];

## SDES lookup tables#

S0_data = [[1, 0, 3, 2],

[3, 2, 1, 0],

[0, 2, 1, 3],

[3, 1, 3, 2]];

S1_data = [[0, 1, 2, 3],

[2, 0, 1, 3],

[3, 0, 1, 0],

[2, 1, 0, 3]];

def ApplyPermutation(X, permutation):r"""This function takes a permutation list (list of bit positions.)And outputs a bit list with the bits taken from X."""

# permute the list Xl = len(permutation);return [X[permutation[j]-1] for j in xrange(l)];

def ApplySBox(X, SBox):r"""This function Applies the SDES SBox (by table look up"""

r = 2*X[0] + X[3];c = 2*X[1] + X[2];o = SBox[r][c];return [o & 2, o & 1];

B.3 / BLock cipherS and the data encryption Standard 675

## Each of these functions uses ApplyPermutation# and a permutation list to perform an SDES# Expansion/Permutation#

def P10(X):return ApplyPermutation(X, P10_data);


def IP(X):return ApplyPermutation(X, IP_data);

def IPinv(X):return ApplyPermutation(X, IPinv_data);

def EP(X):return ApplyPermutation(X, EP_data);


def SW(X):return ApplyPermutation(X, SW_data);

def LS1(X):return ApplyPermutation(X, LS1_data);

def LS2(X):return ApplyPermutation(X, LS2_data);

## These two functions perform the SBox substitutions#

def S0(X):return ApplySBox(X, S0_data);

def S1(X):return ApplySBox(X, S1_data);

def concatenate(left, right):r"""Joins to bit lists together."""ret = [left[j] for j in xrange(len(left))];ret.extend(right);return ret;

def LeftHalfBits(block):r"""Returns the left half bits from block."""


l = len(block);return [block[j] for j in xrange(l/2)];

def RightHalfBits(block):r"""Returns the right half bits from block."""l = len(block);return [block[j] for j in xrange(l/2, l)];

def XorBlock(block1, block2):r"""Xors two blocks together."""l = len(block1);if (l != len(block2)):

raise ValueError, "XorBlock arguments must be same length"

return [(block1[j]+block2[j]) % 2 for j in xrange(l)];

def SDESKeySchedule(K):r"""Expands an SDES Key (bit list) into the two round keys."""temp_K = P10(K);

left_temp_K = LeftHalfBits(temp_K);right_temp_K = RightHalfBits(temp_K);

K1left = LS1(left_temp_K);K1right = LS1(right_temp_K);

K1temp = concatenate(K1left, K1right);K1 = P8(K1temp);

K2left = LS2(K1left);K2right = LS2(K1right);

K2temp = concatenate(K2left, K2right);

K2 = P8(K2temp);

return (K1, K2);

def f_K(block, K):r"""Performs the f_K function supplied block and K."""left_block = LeftHalfBits(block);right_block = RightHalfBits(block);

B.4 / BaSic conceptS in nuMBer theory and Finite FieLdS 677

temp_block1 = EP(right_block);

temp_block2 = XorBlock(temp_block1, K);

left_temp_block2 = LeftHalfBits(temp_block2);right_temp_block2 = RightHalfBits(temp_block2);

S0_out = S0(left_temp_block2);S1_out = S1(right_temp_block2);

temp_block3 = concatenate(S0_out, S1_out);

temp_block4 = P4(temp_block3)

temp_block5 = XorBlock(temp_block4, left_block);

output_block = concatenate(temp_block5, right_block)

return output_block;

def SDESEncrypt(plaintext_block, K):r"""Performs a single SDES plaintext block encryption.(Given plaintext and key as bit lists.)"""

(K1, K2) = SDESKeySchedule(K);

temp_block1 = IP(plaintext_block);

temp_block2 = f_K(temp_block1, K1);

temp_block3 = SW(temp_block2);

temp_block4 = f_K(temp_block3, K2);

output_block = IPinv(temp_block4);

return output_block;

B.4 chapter 4: Basic concepts in nuMBer theory and Finite FieLds

Example 1: The Euclidean algorithm for the greatest common divisor

def EUCLID(a,b):r"""The Euclidean algorithm for finding the gcd of a and b.

This algorithm assumes that a > b => 0

INPUT:

a - positive integer

b - nonnegative integer less than a


OUTPUT:

g - greatest common divisor of a and b"""

if (b < 0) or ( a <= b):raise ValueError, "Expected 0 < a < b"

(A, B) = (a,b);

while (True):

if (0 == B):return A;

R = A % B;A = B;B = R;

Example 2: The extended Euclidean algorithm for the greatest common divisor

def EXTENDED_EUCLID(m,b):r"""The extended Euclidean algorithm to find gcd(m,b).The input is expected to be such that 0 <= b < m.

INPUT:

m - positive integer

b - nonnegative integer less than m

OUTPUT:

(g, b_inv) - g is the gcd of m and b, b_inv is the multiplicative inverse of b mod m.

"""

if (m < b) or (b < 0):

raise ValueError, "Expected input (0 < b < m)"

(A1,A2,A3) = (1,0,m);(B1,B2,B3) = (0,1,b);

while (True):

if (0 == B3):return (A3, None)

if (1 == B3):return (B3, B2)

Q = floor(A3/B3)

(T1,T2,T3) = (A1-Q*B1, A2-Q*B2, A3-Q*B3)


(A1, A2, A3) = (B1, B2, B3)(B1, B2, B3) = (T1, T2, T3)

Example 3: Euclidean algorithm to find gcd of polynomials (with coefficients in a field)

def POLYNOMIAL_EUCLID(A, B):r"""Euclidian algorithm for polynomial GCD:Given two polynomials over the same base field,Assuming degree(A) => degree(B) => 0.

INPUT:

A - polynomial over a field.

B - polynomial over the same field as A, and 0 <= degree(B) <= degree(A).

OUTPUT:

G - greatest common divisor of A and B.

"""

degA = A.degree();degB = B.degree();

if ((degB < 0) or (degA < degB)):raise ValueError, "Expected 0 <= degree(B) <= degree(A)"

while(True):

if (0 == B):return A;

R = A % B;

A = B;B = R;

Example 4: Extended Euclidean algorithm for the gcd of two polynomials (with coefficients in the same field)

def POLYNOMIAL_EXTENDED_EUCLID(m, b):r"""Extended Euclidian algorithm for polynomial GCD:Given two polynomials over the same base field,Assuming degree(m) => degree(b) => 0

INPUT:

m - polynomial over a field.


b - polynomial over the same field as A, and 0 <= degree(B) <= degree(M).

OUTPUT:

(g,b_inv) - the pair where:

g - greatest common divisor of m and b.

m_inv - is None if G is not of degree 0, and otherwise it is the polynomial such that b(X)*b_inv(X) = 1 mod m(X)

"""

degm = m.degree();degb = b.degree();

if(degb < 0) or (degm < degb):raise ValueError, "expected 0 <= degree(b) <= degree(m)"

(A1, A2, A3) = (1, 0, m);(B1, B2, B3) = (0, 1, b);

while (True):

if (0 == B3):return (A3, None);

if (0 == B3.degree()):return (B3/B3, B2/B3);

Q = A3.quo_rem(B3)[0];

(T1, T2, T3) = (A1 - Q*B1, A2 - Q*B2, A3 - Q*B3);(A1, A2, A3) = (B1, B2, B3);(B1, B2, B3) = (T1, T2, T3);

Example 5: Sage has built in functionality for the gcd function. The regular greatest common divisor function can simply be called as:

sage: gcd(15,100)5

sage: gcd(90,65311)1

You can also call it as a method on Integer objects:

sage: x = 10456890sage: x.gcd(100)10

The extended Euclidean algorithm for the greatest common divisor is also built into Sage. Calling xgcd(a,b) returns a tuple, the first element


is the gcd, the second and third elements are coefficients u,v such that gcd(a,b) = u* a + v* b. This can be called as:

sage: xgcd(17,31)(1, 11, -6)sage: xgcd(10, 115)(5, -11, 1)

This can also be called as a method on Integer objects

sage: x = 300sage: x.xgcd(36)(12, 1, -8)

Example 6: Sage includes robust support for working with finite fields and performing finite field arithmetic. To initialize a finite field with prime order, use the GF command passing the order as the parameter.

sage: F = GF(2)sage: FFinite Field of size 2

sage: F = GF(37)sage: FFinite Field of size 37

sage: p = 95131sage: K = GF(p)sage: KFinite Field of size 95131

To initialize a field with a prime power order use the GF command with the following syntax (to keep track of the primitive element of the extension field).

sage: F.<a> = GF(128)sage: FFinite Field in a of size 2^7

To do arithmetic in finite fields use the following syntax:

sage: K = GF(37)sage: a = K(3)sage: b = K(18)sage: a - b22sage: a + b21sage: a * b17sage: a/b31


sage: a^-125sage: 1/a25

To do arithmetic in a finite field with a prime power order, specify ele-ments using the primitive element:

sage: F.<a> = GF(128)sage: b = a^2 + 1sage: c = a^5 + a^3 + 1sage: b - ca^5 + a^3 + a^2sage: b + ca^5 + a^3 + a^2sage: b*ca^3 + a^2 + asage: b/ca^5 + a^3 + a^2 + asage: b^-1a^5 + a^3 + asage: 1/ba^5 + a^3 + a

Example 7: With Sage you can create rings of polynomials over finite fields and do arithmetic with them. To create polynomial rings over finite fields do the following:

sage: R.<x> = GF(2)[]sage: R

Univariate Polynomial Ring in x over Finite Field of size 2 (using NTL)sage: R.<x> = GF(101)[]

sage: Rsage: R.<x> = F[]sage: RUnivariate Polynomial Ring in x over Finite Field in a of size 2^7

After initializing a polynomial ring, you can then just perform arithmetic as you would expect:

sage: R.<x> = GF(2)[]sage: f = x^3 + x + 1sage: g = x^5 + xsage: f + gx^5 + x^3 + 1sage: f*gx^8 + x^6 + x^5 + x^4 + x^2 + x


Division is accomplished by the quo_rem function:

sage: g.quo_rem(f)(x^2 + 1, x^2 + 1)

You can also compute the greatest common divisor:

sage: f.gcd(g)1

sage: g.gcd(g^2)x^5 + x

sage: R.<x> = GF(17)[]sage: f = 3*x^3 + 2*x^2 + xsage: g = x^2 + 5sage: f - g3*x^3 + x^2 + x + 12sage: f * g3*x^5 + 2*x^4 + 16*x^3 + 10*x^2 + 5*xsage: f.quo_rem(g)(3*x + 2, 3*x + 7)

And computing gcds in this polynomial ring we see:

sage: f.gcd(g)1

sage: f.gcd(x^2 + x)x

When creating a Sage finite field with a prime power order, Sage finds an irreducible polynomial for you. For example:

sage: F.<a> = GF(32)a^5 + a^2 + 1

However, there are many irreducible polynomials over GF(2) of degree 5, such as x^5 + x^3 + 1. Suppose that you want to create your own extension of the binary field with degree 5, and an irreducible polynomial of your choice. Then you can do so as follows:

sage: R.<x> = GF(2)[]sage: F = GF(2).extension(x^5 + x^3 + 1, 'a')sage: a = F.gen()

You need to do this last step to inject the primitive element into the interpreter’s name space. This is done automatically when using the GF func-tion to create an extension field, but not when you use the member function extension on a field object.


B.5 chapter 5: advanced encryption standard

Example 1: Simplified AES.

## These structures are the underlying# Galois Field and corresponding Vector Space# of the field used in the SAES algorithm# These structures allow us to easily compute with these fields.

#F = GF(2);L.<a> = GF(2^4);V = L.vector_space();VF8 = VectorSpace(F, 8);

## The MixColumns and its Inverse matrices are stored# as 2x2 matrices with elements in GF(2^4) (as are state matrices.)# The MixColumns operation (and its inverse) are performed by# matrix multiplication.#

MixColumns_matrix = Matrix(L, [[1,a^2],[a^2,1]]);

InverseMixColumns_matrix = MixColumns_matrix.inverse();

SBox_matrix = Matrix(L,[[ 1 + a^3, a^2, a + a^3, 1 + a + a^3],[ 1 + a^2 + a^3, 1, a^3, 1 + a^2],[ a + a^2, 0, a, 1 + a],[ a^2 + a^3, a + a^2 + a^3, 1 + a + a^2 + a^3, 1 + a + a^2]]);

InverseSBox_matrix = Matrix(L,[[ a + a^3, 1 + a^2, 1 + a^3, 1 + a + a^3],[ 1, 1 + a + a^2, a^3, 1 + a + a^2 + a^3],[ a + a^2, 0, a, 1 + a],[ a^2 + a^3, a^2, 1 + a^2 + a^3, a + a^2 + a^3]]);

RCON = [VF8([F(0), F(0), F(0), F(0), F(0), F(0), F(0), F(1)]),VF8([F(0), F(0), F(0), F(0), F(1), F(1), F(0), F(0)])];

B.5 / advanced encryption Standard 685

def SAES_ToStateMatrix(block):r"""Converts a bit list into an SAES state matrix."""

B = block;

# form the plaintext block into a matrix of GF(2^n) elements

S00 = L(V([B[0], B[1], B[2], B[3]]));

S01 = L(V([B[4], B[5], B[6], B[7]]));

S10 = L(V([B[8], B[9], B[10], B[11]]));

S11 = L(V([B[12], B[13], B[14], B[15]]));

state_matrix = Matrix(L, [[S00,S01],[S10,S11]]);

return state_matrix;

def SAES_FromStateMatrix(State Matrix):r"""Converts an SAES State Matrix to a bit list."""

output = [];

# convert State Matrix back into bit listfor r in xrange(2):

for c in xrange(2):v = V(State Matrix[r,c]);for j in xrange(4):

output.append(Integer(v[j]));

return output;

def SAES_AddRoundKey(state_matrix, K):r"""Adds a round key to an SAES state matrix."""

K_matrix = SAES_ToStateMatrix(K);

next_state_matrix = K_matrix + state_matrix;

return next_state_matrix;

def SAES_MixColumns(state_matrix):r"""Performs the Mix Columns operation."""

next_state_matrix = MixColumns_matrix*state_matrix;return next_state_matrix;


def SAES_InverseMixColumns(state_matrix):r"""Performs the Inverse Mix Columns operation."""

next_state_matrix = InverseMixColumns_matrix* state_matrix;return next_state_matrix;

def SAES_ShiftRow(state_matrix):r"""Performs the Shift Row operation."""

M = state_matrix;next_state_matrix = Matrix(L, [

[M[0,0], M[0,1]],[M[1,1], M[1,0]]]);


def SAES_SBox(nibble):r"""Performs the SAES SBox look up in the SBox matrix (lookup table.)"""

v = nibble._vector_();c = Integer(v[0]) + 2*Integer(v[1]);r = Integer(v[2]) + 2*Integer(v[3]);return SBox_matrix[r,c];

def SAES_NibbleSubstitution(state_matrix):r"""Performs the SAES SBox on each element of an SAES state matrix."""

M = state_matrix;next_state_matrix = Matrix(L,

[ [ SAES_SBox(M[0,0]), SAES_SBox(M[0,1])],[ SAES_SBox(M[1,0]), SAES_SBox(M[1,1])] ]);


def SAES_InvSBox(nibble):r"""Performs the SAES Inverse SBox look up in the SBox matrix (lookup table.)"""

v = nibble._vector_();c = Integer(v[0]) + 2*Integer(v[1]);

B.5 / advanced encryption Standard 687

r = Integer(v[2]) + 2*Integer(v[3]);return InverseSBox_matrix[r,c];

def SAES_InvNibbleSub(state_matrix):r"""Performs the SAES Inverse SBox on each element of an SAES state matrix."""

M = state_matrix;next_state_matrix = Matrix(L,[ [ SAES_InvSBox(M[0,0]), SAES_InvSBox(M[0,1])],[ SAES_InvSBox(M[1,0]), SAES_InvSBox(M[1,1])] ]);


def RotNib(w):r"""Splits an 8 bit list into two elements of GF(2^4)"""N_0 = L(V([w[j] for j in xrange(4)]));N_1 = L(V([w[j] for j in xrange(4,8)]));return (N_1, N_0);

def SAES_g(w, i):r"""Performs the SAES g function on the 8 bit list w."""(N0, N1) = RotNib(w);N0 = V(SAES_SBox(N0));N1 = V(SAES_SBox(N1));temp1 = VF8( [ N0[0], N0[1], N0[2], N0[3],

N1[0], N1[1], N1[2], N1[3] ] );output = temp1 + RCON[i];return output;

def SAES_KeyExpansion(K):r"""Expands an SAES key into two round keys."""w0 = VF8([K[j] for j in xrange(8)]);w1 = VF8([K[j] for j in xrange(8,16)]);

w2 = w0 + SAES_g(w1, 0);w3 = w1 + w2;

w4 = w2 + SAES_g(w3, 1);w5 = w3 + w4;

K0 = [w0[j] for j in xrange(8)];K0.extend([w1[j] for j in xrange(8)]);




return (K0, K1, K2);

## Encrypts one plaintext block with key K#

def SAES_Encrypt(plaintext, K):r"""Performs a SAES encryption on a single plaintext block.(Both block and key passed as bit lists.)"""

# get the key schedule(K0, K1, K2) = SAES_KeyExpansion(K);

state_matrix0 = SAES_ToStateMatrix(plaintext);

state_matrix1 = SAES_AddRoundKey(state_matrix0, K0);

state_matrix2 = SAES_NibbleSubstitution (state_matrix1);

state_matrix3 = SAES_ShiftRow(state_matrix2);

state_matrix4 = SAES_MixColumns(state_matrix3);


state_matrix6 = SAES_NibbleSubstitution (state_matrix5);



output = SAES_FromStateMatrix(state_matrix8);

return output;

## Decrypts one ciphertext block with key K#

def SAES_Decrypt(ciphertext, K):r"""Performs a single SAES decryption operation on a ciphertext block.(Both block and key passed as bit lists.)"""

# perform key expansion(K0, K1, K2) = SAES_KeyExpansion(K);

B.6 / pSeudorandoM nuMBer generation and StreaM cipherS 689

# form the ciphertext block into a matrix of GF(2^n) elements

state_matrix0 = SAES_ToStateMatrix(ciphertext);



state_matrix3 = SAES_InvNibbleSub(state_matrix2);


state_matrix5 = SAES_InverseMixColumns (state_matrix4);


state_matrix7 = SAES_InvNibbleSub(state_matrix6);


output = SAES_FromStateMatrix(state_matrix8);

return output;

B.6 chapter 6: pseudorandoM nuMBer generation and streaM ciphers

Example 1: Blum Blum Shub RNG.

def BlumBlumShub_Initialize(bitlen, seed):r"""Initializes a Blum-Blum-Shub RNG State.

A BBS-RNG State is a list with two elements:[N, X]N is a 2*bitlen modulus (product of two primes)X is the current state of the PRNG.

INPUT:

bitlen - the bit length of each of the prime factors of n

seed - a large random integer to start out the prng

OUTPUT:

state - a BBS-RNG internal state

"""

# note that this is not the most cryptographically secure


# way to generate primes, because we do not know how the# internal sage random_prime function works.

p = 3;while (p < 2^(bitlen-1)) or (3 != (p % 4)):

p = random_prime(2^bitlen);

q = 3;while (q < 2^(bitlen-1)) or (3 != (q % 4)):

q = random_prime(2^bitlen);

N = p*q;

X = (seed^2 % N)

state = [N, X]

return state;

def BlumBlumShub_Generate(num_bits, state):r"""Blum-Blum-Shum random number generation function.

INPUT:

num_bits - the number of bits (iterations) to generate with this RNG.

state - an internal state of the BBS-RNG (a list [N, X].)

OUTPUT:

random_bits - a num_bits length list of random bits.

"""

random_bits = [];

N = state[0]X = state[1]

for j in xrange(num_bits):

X = X^2 % N

random_bits.append(X % 2)

# update the internal statestate[1] = X;

return random_bits;

Example 2: Linear Congruential RNG.

def LinearCongruential_Initialize(a, c, m, X0):

B.7 / nuMBer theory 691

r"""This functional initializes a linear congruential RNG state.

This state is a list of four integers: [a, c, m, X]

a,c,m are the parameters of the linear congruential instantiation X is the current state of the PRNG.

INPUT:

a - The coefficientc - The offsetm - The modulusX0 - The initial state

OUTPUT:

state - The initial internal state of the RNG

"""

return [a,c,m,X0]

def LinearCongruential_Generate(state):r"""Generates a single linear congruential RNG output and updates the state.

INPUT:

state - an internal RNG state.

OUTPUT:

X - a single output of the linear congruential RNG.

"""

a = state[0]c = state[1]m = state[2]X = state[3]X_next = (a*X + c) % mstate[3] = X_nextreturn X_next

B.7 chapter 8: nuMBer theory

Example 1: Chinese Remainder Theorem.

def chinese_remainder_theorem(moduli, residues):r"""


Function that implements the chinese remainder theorem.

INPUT:

moduli - list or positive integers.

residues - list of remainders such that remainder at position j results when divided by the correspond-ing modulus at position j in moduli.

OUTPUT:

x - integer such that division by moduli[j] gives remainder residue[j].

"""

if (len(moduli) != len(residues)):

raise ValueError, "expected len(moduli) == len(residues)"

M = prod(moduli);

x = 0;

for j in xrange(len(moduli)):Mj = moduli[j]Mpr = M/Mj

(Mj_Mpr_gcd, Mpr_inv, Mj_inv) = xgcd(Mpr, Mj)

Mpr_inv = Mpr_inv

if (Mj_Mpr_gcd != 1):

raise ValueError, "Expected all moduli are coprime."

x += residues[j]*Mpr*Mpr_inv;

return x;

Example 2: Miller Rabin Primality Test.

r"""

EXAMPLES:

sage: MILLER_RABIN_TEST(101)False

sage: MILLER_RABIN_TEST(592701729979)True

"""


def MILLER_RABIN_TEST(n):

r"""

This function implements the Miller-Rabin Test.It either returns "inconclusive" or "composite."

INPUT:

n - positive integer to probabilistically determine the primality of.

OUTPUT:

If the function returns False, then the test was inconclusive.

If the function returns True, then the test was conclusive and n is composite.

"""

R = IntegerModRing(n); # object for integers mod n# (1) Find integers k, q w/ k > 0 and q odd so that (n-1) == 2^k * qq = n-1k = 0while (1 == (q % 2)):

k += 1q = q.quo_rem(2)[0] # q/2 but with result of type Integer

# (2) select random a in 1 < a < n-1

a = randint(1,n-1)

a = R(a) # makes it so modular exponentiation is done fast

# if a^q mod n == 1 then return inconclusiveif (1 == a^q):

return False

# (3) for j = 0 to k-1 do: if a^(2^j * q) mod n = n-1 return inconclusivee = q

for j in xrange(k):if (n-1) == (aê):return Falsee = 2*e

# (4) if you’ve made it here return composite.return True


Example 3: Modular Exponentiation (Square and Multiply).

def ModExp(x,e,N):r"""Calculates xê mod N using square and multiply.

INPUT:

x - an integer.e - a nonnegative integer.N - a positive integer modulus.

OUTPUT:

y - xê mod N

"""

e_bits = e.bits()

e_bitlen = len(e_bits)

y = 1

for j in xrange(e_bitlen):

y = y^2 % N

if (1 == e_bits[e_bitlen-1-j]):y = x*y % N

return y

Example 4: Using built-in Sage functionality for CRT.Sage has built in functions to perform the Chinese Remainder Theorem.

There are several functions that produce a wide array of CRT functionality. The simplest function performs the CRT with two modulii. Specifically CRT (or the lowercase crt) when called as:

crt(a,b,m,n)

will return a number that is simultaneously congruent to a mod m and b mod n. All parameters are assumed to be integers and the parameters m, n must be relatively prime. Some examples of this function are:

sage: CRT(8, 16, 17, 49)-3120

sage: CRT(1,2,5,7)16

sage: CRT(50,64,101,127)-62166

If you want to perform the CRT with a list of residues and moduli, Sage includes the function CRT_list.


CRT_list(v, modulii)

requires that v and modulii be lists of integers of the same length. Furthermore, the elements of modulii must be relatively prime. Then the output is an integer that reduces to v[i] mod modulii[i] (for i in range(len(v))). For example, the last call to CRT would have been

sage: CRT_list([50,64],[101,127])1969

Note that this answer is different. However, you can check that both answers satisfy the requirements of the CRT. Here are examples with longer lists:

sage: CRT_list([8, 20, 13], [49, 101, 127])608343

sage: CRT_list([10,11,12,13,14],[29,31,37,41,43])36657170

The function CRT_basis can be used to precompute the values associated to the given set of modulii. If modulii is a list of relatively prime modulii, then CRT_basis(modulii) returns a list a. This list a is such that if x is a list of resi-dues of the modulii, then the output of the CRT can be found by summing:

a[0]*x[0] + a[1]*x[1] + ... + a[len(a)-1]*x[len(a)-1]

In the case of the modulii used in the last call to CRT_list this function returns as follows:

sage: CRT_basis([29,31,37,41,43])[32354576, 20808689, 23774055, 17163708, 23184311]

The last CRT function that Sage provides is CRT_vectors. This function performs CRT_list on several different lists (with the same set of modulii) and returns a list of the simultaneous answers. It is efficient in that it uses CRT_basis and does not recompute those values for each list. For example:

sage: CRT_vectors([[1,10],[2,11],[3,12],[4,13],[5,14]], [29,31,37,41,43])[36657161, 36657170]

Example 5: Using built-in Sage functionality for Modular Exponentiation.Sage can perform modular exponentiation using fast algorithms (like

square and multiply) and without allowing the intermediate computations to become huge. This is done through IntegerModRing objects. Specifically, cre-ating an IntegerModRing object indicates that arithmetic should be done with a modulus. Then you cast your integers in this ring to indicate that all arithme-tic should be done with the modulus. Then for elements of this ring, exponen-tiation is done efficiently. For example:

sage: R = IntegerModRing(101)


sage: x = R(10)sage: x^9991

sage: R = IntegerModRing(1024)sage: x = R(111)sage: x^345751

sage: x = R(100)sage: x^2000

sage: N = 127*101sage: R = IntegerModRing(N)sage: x = R(54)sage: x^959177

Creating an IntegerModRing is similar to creating a FiniteField with GF(...) except that the modulus can be a general composite.

Example 6: Using built-in Sage functionality for Euler’s totient.Sage has the Euler totient functionality built in. The function is called

euler_phi because of the convention of using the Greek letter phi to represent this function. The operation of this function is simple. Just call euler_phi on an integer and it computes the totient function. This function factors the input, and hence requires exponential time.

sage: euler_phi(101)100





B.8 chapter 9: puBLic-key cryptography and rsa

Example 1: Using Sage we can simulate an RSA encryption and decryption.

sage: # randomly select some prime numberssage: p = random_prime(1000); p191

B.8 / puBLic-key cryptography and rSa 697

sage: q = random_prime(1000); q601sage: # compute the modulussage: N = p*qsage: R = IntegerModRing(N)sage: phi_N = (p-1)*(q-1)sage: # we can choose the encrypt key to be anythingsage: # relatively prime to phi_Nsage: e = 17sage: gcd(d, phi_N)1sage: # the decrypt key is the multiplicative inversesage: # of d mod phi_Nsage: d = xgcd(d, phi_N)[1] % phi_Nsage: d60353sage: # Now we will encrypt/decrypt some random 7 digit numbers

sage: P = randint(1,127); P97sage: # encryptsage: C = R(P)ê; C46685sage: # decryptsage: R(C)^d97



Also, Sage can just as easily do much larger numbers:

sage: p = random_prime(1000000000); p


114750751sage: q = random_prime(1000000000); q8916569sage: N = p*qsage: R = IntegerModRing(N)sage: phi_N = (p-1)*(q-1)sage: e = 2^16 + 1sage: d = xgcd(e, phi_N)[1] % phi_Nsage: d237150735093473

sage: P = randint(1,1000000); P955802sage: C = R(P)êsage: R(C)^d955802

Example 2: In Sage, we can also see an example of RSA signing/verifying.

sage: p = random_prime(10000); p1601sage: q = random_prime(10000); q4073sage: N = p*qsage: R = IntegerModRing(N)sage: phi_N = (p-1)*(q-1)sage: e = 47sage: gcd(e, phi_N)1sage: d = xgcd(e,phi_N)[1] % phi_Nsage: # Now by exponentiating with the private keysage: # we are effectively signing the datasage: # a few examples of this

sage: to_sign = randint(2,2^10); to_sign650sage: # the signature is checked by exponentiatingsage: # and checking vs the to_sign valuesage: signed = R(to_sign)^d; signed2910116sage: to_sign == signedêTruesage: to_sign = randint(2,2^10); to_sign362sage: signed = R(to_sign)^d; signed546132sage: to_sign == signedêTrue

B.9 / other puBLic-key cryptoSySteMS 699

sage: # we can also see what happens if we try to verify a bad signature

sage: to_sign = randint(2,2^10); to_sign605sage: signed = R(to_sign)^d; signed1967793sage: bad_signature = signed - randint(2,100)sage: to_sign == bad_signatureêFalse

B.9 chapter 10: other puBLic-key cryptosysteMs

Example 1: Here is an example of Alice and Bob performing a Diffie-Hellman Key Exchange done in Sage:

sage: # Alice and Bob agree on the domain parameters:sage: p = 619sage: F = GF(p)sage: g = F(2)sage: # Alice picks a random value x in 1...618sage: x = randint(1,618); x571sage: # Alice computes X = g^x and sends this to Bobsage: X = g^571; X591sage: # Bob picks a random value y in 1...618sage: y = randint(1,618);y356sage: # Bob computes Y = g^y and sends this to Alicesage: Y = g^y; Y199sage: # Alice computes Y^xsage: Y^x563sage: # Bob computes X^ysage: X^y563sage: # Alice and Bob now share a secret value

Example 2: In reality to prevent what is known as small subgroup attacks,the prime p is chosen so that p - 2q + 1 where p is a prime as well.

sage: q = 761sage: p = 2*q + 1sage: is_prime(q)True


sage: is_prime(p)Truesage: F = GF(p)sage: g = F(3)sage: g^q1sage: # note that g^q = 1 implies g is of order qsage: # Alice picks a random value x in 2...q-1sage: x = randint(2,q-1); x312sage: # Alice computes X = g^x and sends it to Bobsage: X = g^x; X26sage: # Bob computes a random value y in 2...q-1sage: y = randint(2,q-1); y24sage: # Bob computes Y = g^y and sends it to Alicesage: Y = g^y; Y1304sage: # Alice computes Y^xsage: Y^x541sage: # Bob computes X^ysage: X^y541sage: # Alice and Bob now share the secret value 541

Example 3: Sage has a significant amount of support for elliptic curves. This functionality can be very useful when learning, because it allows you to easily calculate things and get the big picture. Doing the examples by hand may cause you to get mired in the details. First you instantiate an elliptic curve, by speci-fying the field that it is over, and the coefficients of the defining Weierstrass equation. For this purpose, we write the Weierstrass equation as

y2 + a1xy + a3y = x3 + a2x2 + a4x + a6

Then the Sage function EllipticCurve(R, [a1, a2, a3, a4, a6]) creates the elliptic curve over the ring R.

sage: E = EllipticCurve(GF(17), [1,2,3,4,5])sage: EElliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Finite Field of size 17

sage: E = EllipticCurve(GF(29), [0,0,0,1,1])sage: EElliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 29


sage: E = EllipticCurve(GF(127), [0,0,0,2,17])sage: EElliptic Curve defined by y^2 = x^3 + 2*x + 17 over Finite Field of size 127

sage: F.<theta> = GF(2^10)sage: E = EllipticCurve(F, [1,0,0,1,0])sage: EElliptic Curve defined by y^2 + x*y = x^3 + x over Finite Field in theta of size 2^10

Example 4: Koblitz curves. A Koblitz curve is an elliptic curve over a binary field defined by an equation of the form

y2 + xy = x3 + ax2 + 1

where a = 0 or 1. FIPS 186-3 recommends a number of Koblitz curves for use with the Digital Signature Standard (DSS). Here we give an example of a curve of similar form to the Koblitz curves:

sage: F.<theta> = GF(2^17)sage: E = EllipticCurve(F,[1,0,0,theta,1])sage: EElliptic Curve defined by y^2 + y = x^3 + theta* x^2 = 1 over Finite Field in theta of size 2^17

Example 5: Sage can even easily instantiate curves of cryptographic sizes, like K163, which is one of the FIPS 186-3 curves.

sage: F.<theta> = GF(2^163)sage: E = EllipticCurve(F, [1,0,0,1,1])sage: EElliptic Curve defined by y^2 + x*y = x^3 + x^2 + 1 over Finite Field in theta of size 2^163

However, you should be careful that when instantiating a curve of crypto-graphic sizes, some of the functions on the curve object will not work because they require exponential time to run. While you can compute some things with these objects, it is best to leave your experimentation to the smaller sized curves.

You can calculate some values of the curve, such as the number of points:

sage: E = EllipticCurve(GF(107), [0,0,0,1,0])sage: E.order()108

You can also determine the generators of a curve:

sage: E = EllipticCurve(GF(101), [0,0,0,1,0])sage: E.gens()((7 : 42 : 1), (36 : 38 : 1))


Note that this output is printed (x : y : z). This is a minor technical consideration because Sage stores points in what is known as “projective coordinates.” The precise meaning is not important, because for non-infinite points the value z will always be 1 and the first two values in a coordinate will be the x and y coordi-nates, exactly as you would expect. This representation is useful because it allows the point at infinity to be specified as a point with the z coordinate equal to 0:

sage: E(0)(0 : 1 : 0)

This shows how you can recognize a point at infinity as well as specify it. If you want to get the x and y coordinates out of a point on the curve, you can do so as follows:

sage: P = E.random_point(); P(62 : 38 : 1)sage: (x,y) = P.xy(); (x,y)(62, 38)

You can specify a point on the curve by casting an ordered pair to the curve as:

sage: P = E((62,-38)); P(62 : 63 : 1)

Now that you can find the generators on a curve and specify points you can experiment with these points and do arithmetic as well. Continuing to use E as the curve instantiated in the previous example, we can set G1 and G2 to the generators:

sage: (G1, G2) = E.gens()sage: P = E.random_point(); P(49 : 29 : 1)

You can compute the sum of two points as in the following examples:

sage: G1 + G2 + P(69 : 96 : 1)sage: G1 + P(40 : 62 : 1)sage: P + P + G2(84 : 25 : 1)

You can compute the inverse of a point using the unary minus (−) operator:

sage: -P(49 : 72 : 1)sage: -G1(7 : 59 : 1)

You can also compute repeated point addition (adding a point to itself many times) with the * operator:

sage: 13*G1(72 : 23 : 1)


sage: 2*G2(9 : 58 : 1)sage: 88*P(87 : 75 : 1)

And for curves over small finite fields you can also compute the order (discrete log of the point at infinity with respect to that point).

sage: G1.order()10

sage: G2.order()10

sage: P.order()10

Example 6: Using the Sage elliptic curve functionality to perform a simulated elliptic curve Diffie-Hellman (ECDH) key exchange.

sage: # calculate domain parameterssage: F = GF(127)sage: E = EllipticCurve(F, [0, 0, 0, 3, 4])sage: G = E.gen(0); G(94 : 6 : 1)sage: q = E.order(); q122

sage: # Alice computes a secret value x in 2 ... q-1sage: x = randint(2,q-1); x33

sage: # Alice computes a public value X = x*Gsage: X = x*G; X(55 : 89 : 1)

sage: # Bob computes a secret value y in 2 ... q-1sage: y = randint(2,q-1); y55

sage: # Bob computes a public value Y = y*Gsage: Y = y*G; Y(84 : 39 : 1)

sage: # Alice computes the shared valuesage: x*Y(91 : 105 : 1)

sage: # Bob computes the shared valuesage: y*X(91 : 105 : 1)


However, in practice most curves that are used have a prime order:

sage: # Calculate the domain parameterssage: F = GF(101)sage: E = EllipticCurve(F, [0, 0, 0, 25, 7])sage: G = E((97,34))sage: q = E.order()sage: # Alice computes a secret values x in 2...q-1sage: x = randint(2,q-1)sage: # Alice computes a public value X = x*Gsage: X = x*Gsage: # Bob computes a secret value y in 2...q-1sage: y = randint(2,q-1)sage: # Bob computes a public value Y = y*Gsage: Y = y*Gsage: # Alice computes the shared secret valuesage: x*Y(23 : 15 : 1)sage: # Bob computes the shared secret valuesage: y*X(23 : 15 : 1)

B.10 chapter 11: cryptographic hash Functions

Example 1: The following is an example of the MASH hash function in Sage. MASH is a function based on the use of modular arithmetic. It involves use of an RSA-like modulus M, whose bit length affects the security. M should be difficult to factor, and for M of unknown factorization, the security is based in part on the difficulty of extracting modular roots. M also determines the block size for processing messages. In essence, MASH is defined as:

Hi = ((xi ⊕ Hi- 1)2OR Hi- 1) (mod M)

where

A = 0xFF00 c00Hi- 1 = the largest prime less than Mxi = the ith digit of the base M expansion of input n. That is, we express n as a number of base M. Thus:

n = x0 + x1M + x2M2 + c

The following is an example of the MASH hash function in Sage

## This function generates a mash modulus# takes a bit length, and returns a Mash# modulus l or l-1 bits long (if n is odd)

B.10 / cryptographic haSh FunctionS 705

# returns p, q, and the product N#def generate_mash_modulus(l):

m = l.quo_rem(2)[0]

p = 1while (p < 2^(m-1)):p = random_prime(2^m)

q = 1while (q < 2^(m-1)):q = random_prime(2^m)

N = p*qreturn (N, p, q)

## Mash Hash# the value n is the data to be hashed.# the value N is the modulus# Returns the hash value.#

def MASH(n, N):

H = previous_prime(N)

q = n

while (0 != q):(q, a) = q.quo_rem(N)H = ((H+a)^2 + H) % N

return H

The output of these functions running;

sage: data = ZZ(randint(1,2^1000))sage: (N, p, q) = generate_mash_modulus(20)sage: MASH(data, N)220874sage: (N, p, q) = generate_mash_modulus(50)sage: MASH(data, N)455794413217080sage: (N, p, q) = generate_mash_modulus(100)sage: MASH(data, N)268864504538508517754648285037sage: data = ZZ(randint(1,2^1000))sage: MASH(data, N)236862581074736881919296071248


sage: data = ZZ(randint(1,2^1000))sage: MASH(data, N)395463068716770866931052945515

B.11 chapter 13: digitaL signatures

Example 1: Using Sage, we can perform a DSA sign and verify:

sage: # First we generate the domain parameterssage: # Generate a 16 bit prime qsage: q = 1;sage: while (q < 2^15): q = random_prime(2^16)....:sage: q42697sage: # Generate a 64 bit p, such that q divides (p-1)sage: p = 1sage: while (not is_prime(p)):....: p = (2^48 + randint(1,2^46)*2)*q + 1....:sage: p12797003281321319017sage: # Generate h and gsage: h = randint(2,p-2)sage: h5751574539220326847sage: F = GF(p)sage: g = F(h)^((p-1)/q)sage: g9670562682258945855

sage: # Generate a user public / private keysage: # private keysage: x = randint(2,q-1)sage: x20499sage: # public keysage: y = F(g)^xsage: y7955052828197610751

sage: # Sign and verify a random valuesage: H = randint(2,p-1)sage: # Signingsage: # random blinding value

B.11 / digitaL SignatureS 707

sage: k = randint(2,q-1)sage: r = F(g)^k % qsage: r = F(g)^ksage: r = r.lift() % qsage: r6805sage: kinv = xgcd(k,q)[1] % qsage: s = kinv*(H + x*r) % qsage: s26026

sage: # Verifyingsage: w = xgcd(s,q)[1]; w12250sage: u1 = H*w % q; u16694sage: u2 = r*w % q; u216706sage: v = F(g)û1 * F(y)û2sage: v = v.lift() % qsage: v6805sage: v == rTrue

sage: # Sign and verify another random valuesage: H = randint(2,p-1)sage: k = randint(2,q-1)sage: r = F(g)^ksage: r = r.lift() % qsage: r3284sage: kinv = xgcd(k,q)[1] % qsage: s = kinv*(H + x*r) % qsage: s2330

sage: # Verifyingsage: w = xgcd(s,q)[1]; w4343sage: u1 = H*w % q; u132191sage: u2 = r*w % q; u21614sage: v = F(g)û1 * F(y)û2sage: v = v.lift() % qsage: v


3284sage: v == rTrue

Example 2: The following functions implement DSA domain parameter gen-eration, key generation, and DSA Signing:

## Generates a 16 bit q and 64 bit p, both prime# such that q divides p-1#def DSA_generate_domain_parameters():

g = 1

while (1 == g):

# first find a qq = 1while (q < 2^15): q = random_prime(2^16)# next find a pp = 1while (not is_prime(p)):

p = (2^47 + randint(1,2^45)*2)*q + 1

F = GF(p)

h = randint(2,p-1)

g = (F(h)^((p-1)/q)).lift()

return (p, q, g)

## Generates a users private and public key# given domain parameters p, q, and g#def DSA_generate_keypair(p, q, g):

x = randint(2,q-1)

F = GF(p)

y = F(g)^xy = y.lift()

return (x,y)

## Given domain parameters p, q and g# as well as a secret key x# and a hash value H# this performs the DSA signing algorithm#

B.11 / digitaL SignatureS 709

def DSA_sign(p, q, g, x, H):

k = randint(2,q-1)

F = GF(p)

r = F(g)^k

r = r.lift() % q

kinv = xgcd(k,q)[1] % q

s = kinv*(H + x*r) % q

return (r, s)

710710

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Pages 563–566: Souppaya, M. and Scarfone, K.: SP 800-124 (Guidelines for Managing and Securing Mobile Devices in the Enterprise) NIST.Page 574: IEEE 802.11™ WIRELESS LOCAL AREA NET-WORKS The Working Group for WLAN Standards, IEEE.Page 574: Establishing Wireless Robust Security Networks: A Guide to IEEE 802.11i, NIST.Page 575: IEEE 802.11™ WIRELESS LOCAL AREA NET-WORKS The Working Group for WLAN Standards, IEEE.Page 579: Frankel, S.; Eydt, B.; Owens, L.; and Scarfone, K. “Establishing Wireless Robust Security Networks: A Guide to IEEE 802.11i.” NIST Special Publication SP 800-97, February 2007.Page 580: Frankel, S.; Eydt, B.; Owens, L.; and Scarfone, K. “Establishing Wireless Robust Security Networks: A Guide to IEEE 802.11i.” NIST Special Publication SP 800-97, February 2007.Page 586: Frankel, S.; Eydt, B.; Owens, L.; and Scarfone, K. “Establishing Wireless Robust Security Networks: A Guide to IEEE 802.11i.” NIST Special Publication SP 800-97, February 2007.Page 591: The Tower Commission Report to President Reagan on the Iran-Contra Affair, 1987.Page 597: Held, G. Data and Image Compression: Tools and Techniques. New York: Wiley, 1996.Page 599: Resnick, P Ed: RFC 5322 - Internet Message Format, Internet Engineering Task Force.Page 600: These materials have been reproduced by Pearson Education, Inc. with the permission of International Business Machines Corporation from IBM Redbooks® publication GG24-3376: TCP/IP Tutorial and Technical Overview (http://www.redbooks.ibm.com/abstracts/gg243376.html?Open”). COPYRIGHT © 2006 International Business Machines Corporation. ALL RIGHTS RESERVED.Page 601: Freed, N. and Borenstein, N: RFC 2046: “Multipurpose Internet Mail Extensions - (MIME) Part Two: Media Types,” The Internet Society.Page 601: Freed, N. and Borenstein, N: RFC 2046: “Multipurpose Internet Mail Extensions - (MIME) Part Two: Media Types,” The Internet Society.Page 607: Bradner: RFC 2119 (Key Words for use in RFCs to Indicate Requirement Levels), The Internet Society.Page 617: Fenton, J: RFC 4686: (Analysis of Threats Motivating DomainKeys Identified Mail), The internet Society, 2006.Page 627: “The Art of War,” Sun Tzu, translated by Lionel Giles, The Project Gutenberg Literary Archive Foundation, May 1994.Page 630: Huitema, C. IPv6: The New Internet Protocol. Upper Saddle River, NJ: Prentice Hall, 1998.Page 630: Kent, S. and Seo, K: “RFC 4301: Security Architecture for the Internet Protocol,” The Internet Society, 2005.Page 630: Huitema, Christian, IPv6: The New Internet Protocol, 2nd Ed., ©1998. Reprinted and Electronically reproduced by permission of Pearon Education, Inc., Upper Saddle River, NJ.Page 662: von Clausewitz, C. “On War,” Lake City, Utah. The Project Gutenberg Literary Archive Foundation, 2006.Page 668–709: Provided by Dan Shumow.Page 710: von Clausewitz, C. “On War,” Lake City, Utah. The Project Gutenberg Literary Archive Foundation, 2006.



http://www.redbooks.ibm.com/abstracts/gg243376.html?Open

http://www.redbooks.ibm.com/abstracts/gg243376.html?Open

AAbelian groups, 101, 103, 295–296Absorbing phase, 341–342Access control, 18, 20, 505, 568–569,

578–579Access point (AP), 560, 562, 569Access requestor (AR), 496Accountability, 10Active threat, 16–17, 25

denial of service, 16masquerade, 16modification of messages, 16replay, 16

Addition, 102, 119, 296–298AddRoundKey, 132, 135

forward add round key transforma-tion, 147

inputs for single AES round, 148inverse add round key transforma-

tion, 147Administrative management domain

(ADMD), 616–617, 620Advanced Encryption Standard (AES),

62, 72–74, 86, 130, 684–689AddRoundKey and

InvMixColumns, 156avalanche effect, 152–155byte-level operations, 138data structures, 134detailed structure, 135–137

AddRoundKey, 135MixColumns, 135ShiftRows, 135substitute bytes, 135

encryptionand decryption, 136process, 133round, 137

equivalent inverse cipher, 155–157general structure, 132–135

State, 132implementation, 155–159

8-bit processor, 15732-bit processor, 158

InvShiftRows and InvSubBytes, 156key expansion, 148–151parameters, 135row and column operations, 144State array, 132

Alert Protocol, 530–531Algorithm, 240–241, 288–290

design, 209–210asymmetric ciphers, 210cryptographic algorithms, 209hash functions and MAC, 210purpose-built algorithms, 209symmetric block ciphers, 210

negotiation, 548ANSI X9.17 PRNG, 215–216

input, 215keys, 215output, 216

Anti-replay service, 640–641receiver, 640replay attack, 640sender, 640

Arbitrary reversible substitution cipher, 66

Associative group, 99, 295Associativity of multiplication, 101Asymmetric card authentication key, 488Asymmetric ciphers, 210Asymmetric encryption

keys, 255

public key certificate, 255public key cryptographic algorithm, 255public key infrastructure (PKI), 255

Authenticated encryption, 376–382CCM, 376–379E&M: Encrypt-and-MAC, 376EtM: Encrypt-then-MAC, 376GCM, 379–382HtE: Hash-then-encrypt, 376MtE: MAC-then-encrypt, 376

Authentication, 454exchange, 21payload, 656protocols, 8server, 501service exchange, 473

Authentication Header (AH), 630–631Authenticator, 261, 358, 501, 504Authenticity, 10Autokey system, 46Availability, 10, 12Availability service, 19–20Avalanche effect, 75–76, 152–155

BBackward unpredictability, 208Base64 transfer encoding, 604Basic service set (BSS), 567, 569–570Big-endian, 331Bijection, 243Bill Bryant of Project Athena

(BRYA88), 460Binary curve, 299BIO, 489BIO-A, 489Birthday attack, 328–329Birthday paradox, 325Bit independence criterion (BIC),

79–8056-Bit keys, 778-Bit processor, 157–15832-Bit processor, 158–159Bitrate, 339, 341–342Block size, 70, 200Blum Blum Shub Generator, 212–213Brute force, 272Brute-force attacks, 31, 33, 325–327

birthday paradox, 325collision resistant attacks, 325preimage and second preimage

attacks, 325Business continuity and disaster

recovery, 520

CCaesar cipher, 34–36, 44–45Capacity, 341–343Card authentication key, 489Cardholder unique identifier (CHUID),

485, 487CBC-MAC or CMAC, 225Certificate payload, 656Certificate request payload, 656Change Cipher Spec Protocol, 530–531Character marking, 53Chinese remainder theorem

bijection, 243first assertion, 243second assertion, 243

Chi step function, 349–350Chosen ciphertext attack (CCA), 33,

272, 277Chosen-plaintext attack, 33Chosen text attack, 33

CIA triad, 10accountability, 10authenticity, 10availability, 10, 12confidentiality, 10–11high level, 11integrity, 10, 12low level, 11moderate level, 11

Cipher, 28block, 31, 63, 673–677

modes of operation, 181projects, 664

design principles, 78–80bit independence criterion (BIC), 79design of function F, 79key schedule algorithm, 80number of rounds, 79strict avalanche criterion (SAC), 79

Cipher-based message authentication code (CMAC), 374–375

Cipher block chaining mode (CBC), 183–185mode, 183nonce, 184

Cipher feedback mode (CFB), 185–187counter (CTR) mode, 185encryption, 185output feedback (OFB) mode, 185s-bit mode, 186segments, 185

Ciphertext, 28–29, 258Ciphertext only attack, 31–32, 33Ciphertext-stealing, 196Classical encryption, 27–54, 670–673Client/server authentication exchange, 473Closure, 99, 295

under multiplication, 101Cloud, 506Cloud auditor, 510, 511–512Cloud broker, 511

service aggregation, 511service arbitrage, 511service intermediation, 511

Cloud carrier, 510, 511Cloud computing, 505–512

characteristics of, 506–508broad network access, 506–507measured service, 507on-demand self-service, 507rapid elasticity, 507resource pooling, 507–508

context, 509deployment models, 508–509

community cloud, 508hybrid cloud, 509private cloud, 508public cloud, 508

elements, 506–509reference architecture, 510–512

cloud auditor, 511cloud broker, 511cloud carrier, 511cloud consumer, 510cloud provider, 511

service models, 508IaaS, 508PaaS, 508SaaS, 508

Cloud consumer, 510, 511Cloud provider, 510, 511Cloud security

risks and countermeasures, 512–513abuse and nefarious use, 512account or service hijacking, 513

Index

723

724 Index

Cloud security (Continued)data loss or leakage, 513insecure interfaces and APIs, 512malicious insiders, 512–513shared technology issues, 513unknown risk profile, 513

as service, 517–520Coefficient set, 106Collision, 322–323Community cloud, 508Commutative, 100–101, 295

ring, 103Composite number, 241Composition, 165, 344Compression function, 327Computational aspects, 267–272Computationally secure encryption

scheme, 33Computer security

availability, 10challenges, 12–13confidentiality

data, 9privacy, 10

definition of, 9integrity, 10

Confidentiality, 10–11Configuration payload, 657Confusion, 67–68Congruent modulo, 91Connection Protocol, 551–555

channel mechanism, 551–552close a channel, 552data transfer, 552open a new channel, 552

channel types, 552direct-tcpip, 552forwarded-tcpip, 552session, 552x11, 552

message exchange, 553port forwarding, 553–555

Consistency, 207Constant polynomial, 106Conventional encryption, 31, 259Cookie exchange, 651Counter Mode-CBC MAC Protocol

(CCMP), 584Counter mode (CTR), 189

hardware efficiency, 190preprocessing, 191provable security, 191random access, 191simplicity, 191software efficiency, 191

Counter with cipher block chaining-message authentication code (CCM), 376–379

CREATE_CHILD_SA exchange, 653Cryptanalysis, 28, 31, 327–328

and brute-force attack, 31, 33computationally secure encryption

scheme, 33types of attacks on encrypted

messages, 32unconditionally secure encryption

scheme, 33compression function, 327structure of secure hash code, 327

Cryptographic algorithms and protocols, 8asymmetric encryption, 8authentication protocols, 8data integrity algorithms, 8symmetric encryption, 8

Cryptographically secure pseudorandom bit generator (CSPRBG), 212

Cryptographic checksum, 362Cryptographic hash functions,

314–320, 323, 387, 704–706collision resistant, 323

digital signatures, 319intrusion detection, 320message authentication, 315–319one-way password file, 320preimage resistant, 323PRF, 320PRNG, 320properties, relationship, 323pseudorandomness, 324requirements and security, 322–328

brute-force attacks, 325–327collision, 322–323cryptanalysis, 327–328for cryptographic hash

functions, 323preimage, 322for public-key cryptography,

262–264for secure use of conventional

encryption, 29resistance properties, 324second preimage resistant, 323virus detection, 320

Cryptographic suites, 657–659Cryptographic system, 28Cryptography, 28, 31

and network securityblock cipher projects, 664case studies, 666firewall projects, 666hacking project, 663–664laboratory exercises, 664practical security assessments,

665–666programming projects, 665reading/report assignments, 667research projects, 664–665sage computer algebra projects,

662–663writing assignments, 666–667

number of keys used, 31conventional encryption, 31secret-key, 31single-key, 31symmetric, 31

plaintext, 31block cipher, 31stream cipher, 31

transforming plaintext to ciphertext, 31

product systems, 31Cryptology, 28CTR_DRBG, 214, 216–218Cubic equation, 296Cyclic group, 100

DData authentication algorithm (DAA),

373–374Data confidentiality, 9, 19Data consumers, 479Data encryption algorithm (DEA), 72Data Encryption Standard (DES), 52, 64,

72–76, 254, 673–677avalanche effect, 75–76decryption, 74double, 175–177

meet-in-the-middle attack, 177multiple encryption, 176reduction to a single stage, 176–177

encryption, 73–74permuted input, 74preoutput, 74strength of, 77–78

nature of the DES algorithm, 77–78timing attacks, 78use of 56-Bit keys, 77

subkey, 74–75Data integrity, 8, 19–20Data loss prevention (DLP), 519

Data protection in the cloud, 514–517attributes, 516entities

client, 516data owner, 516server, 516user, 516

multi-instance model, 515multi-tenant model, 515primary key, 516relation, 516tuples, 516

Deciphering, 28Decryption, 28, 262

algorithm, 29, 258Delete payload, 657Denial of service, 16Deskewing, 224Detached signature, 595Deterministic primality algorithm, 242Diffie-Hellman (DH)

key exchange, 287–291, 304algorithm, 288–290discrete logarithm, 288key exchange protocols, 290man-in-the-middle attack, 290–291order, 304

values, 652Diffusion, 67Digital random number generator

(DRNG), 224–225hardware architecture, 225–227

CBC-MAC or CMAC, 225Intel DRNG logical structure, 226Intel processor chip, 225

logical structure, 227Digital signatures, 20, 260, 262, 319,

395–398, 706–709attacks and forgeries

adaptive chosen message attack, 397directed chosen message attack, 397existential forgery, 397generic chosen message attack, 397key-only attack, 397known message attack, 397selective forgery, 397total break, 397universal forgery, 397

direct, 398essential elements, 396generic model of, 395key, 488properties, 395–396requirements, 397–398simplified examples, 319timestamp, 398

Digital signature algorithm (DSA), 292, 394, 400–404

approach, 401–402signing and verifying, 404

Digram, 37, 39–40Direct digital signature, 398Discrete logarithms, 244–249, 288

calculation of, 248–249for modular arithmetic, 246–248powers of an integer, 245–246

primitive root, 246Disk drives, 223Distribution system (DS), 569, 571Distributive laws, 101Divisibility, 87–88Division algorithm, 87–88Divisor, 109DomainKeys Identified Mail (DKIM),

615–622deployment, 619e-mail threats, 617–618

capabilities, 617–618characteristics, 617location, 618

Index 725

functional flow, 620–622Internet mail architecture, 615–617

Mail Delivery Agent (MDA), 616Mail Submission Agent (MSA), 616Message Store (MS), 616Message Transfer Agent (MTA), 616Message User Agent (MUA),

615–616strategy, 618–619

Domain Name System (DNS), 617Dynamic biometrics, 452Dynamic Host Configuration Protocol

(DHCP), 499

EEAP authenticator, 501–502, 506EAP-GPSK (EAP Generalized

Pre-Shared Key), 500EAP-IKEv2, 500EAP method, 500–501EAP over LAN (EAPOL), 503

-EAP packet, 504packet, 504–505

body, 505body length, 505type, 504protocol version, 504

-Start packet, 504EAP pass-through mode, 501–502EAP peer, 501–502, 506EAP-TLS (EAP Transport Layer

Security), 500EAP-TTLS (EAP Tunneled TLS), 500Ease of analysis, 70Electrically erasable programmable

ROM (EEPROM), 485Electronic code book (ECB), 180–182

block cipher modes of operation, 181diffusion, 182error propagation, 182error recovery, 182mode, 182modes of operation, 180overhead, 182security, 182

Electronic facial image, 488Electronic mail security, 590–622

DKIM, 615–622PGP, 591–598S/MIME, 599–615

Elgamal cryptographic system, 292–294Elgamal digital signature scheme, 398–400Elliptic curve arithmetic, 295–303

abelian groups, 295–296associative, 295closure, 295commutative, 295elliptic curve, 296identity element, 295inverse element, 295

over real numbersalgebraic description of addition,

297–298elliptic curves over GF(2m), 301elliptic curves over Zp, 298–301geometric description of addition,

297–298point at infinity or zero point, 296Weierstrass equation, 296

Elliptic curve cryptography (ECC), 303–306

computational effort for cryptanalysis, 306DH key exchange, 304–305

order, 304encryption/decryption, 304–306security of, 306

Elliptic Curve Digital Signature Algorithm (ECDSA), 394, 404–407

generation and authentication, 405–407

global domain parameters, 405key generation, 405signing and verifying, 406

Elliptic curves over GF(2m), 301Elliptic curves over Zp, 298–301E-mail security, 519E&M: Encrypt-and-MAC, 376Encapsulating security payload (ESP)

anti-replay service, 640–641encryption and authentication

algorithms, 639–640, 643format, 638–639information, 634padding, 640protocol operation, 645transport and tunnel modes,

641–644Enciphering, 20, 28Encrypted messages, types of attacks

on, 32chosen ciphertext, 33chosen-plaintext, 33chosen text, 33ciphertext only, 33known plaintext, 32

Encrypted payload, 657Encryption, 28, 262, 519–520

algorithm, 29, 258asymmetric, 8classical, 670–673and decryption tables for

substitution, 65scheme

computationally secure, 33unconditionally secure, 33

symmetric, 8End-to-end encryption, 419Entropy source, 205, 217, 223EtM: Encrypt-then-MAC, 376Euclidean algorithm, 88–91

example, 91greatest common divisor, 88–91relatively prime, 88

Euler’s theorems, 236–239Euler’s totient function, 237Event detection, 20Extended service set (ESS), 570Extensible Authentication Protocol

(EAP), 499–503, 657authentication methods, 500

EAP-GPSK (EAP Generalized Pre-Shared Key), 500

EAP-IKEv2, 500EAP-TLS (EAP Transport Layer

Security), 500EAP-TTLS (EAP Tunneled TLS),

500exchanges, 500–503

fields, 502in pass-through mode, 501–502

-Key packet, 504layered context, 499-Logoff packet, 504messages

code, 502data, 502identifier, 502length, 502

Extensible Markup Language (XML), 482

FFactor, 109Factoring problem, 272–275Family Educational Rights and Privacy

Act (FERPA), 11Fast software encryption/decryption, 70Fault-based attack, 276Federated identity management,

478–484

identity management, 478–479, 480identity operation, 480–484

Feedback characteristic of modes of operation, 192

Feistel cipher, 66–67decryption, 69–72encryption, 69example, 72structure, 68–70

Fermat’s theorems, 236–239Fields, 102–103

multiplicative inverse, 102Finite fields, 301

arithmetic, 130–132of form GF(p), 102–105

multiplicative inverse, 104–105order p, 102–104

irreducible, 131Finite group, 100Finite ring, 101FIPS PUB 199, 11Firewall, 499, 565Firewall projects, 666First assertion, 243Forward add round key transformation

(AddRoundKey), 147Forward mix column transformation

(MixColumns), 144Forward shift row transformation

(ShiftRows), 143Forward substitute byte transformation

(SubBytes), 138Forward unpredictability, 208Frequency test, 208

GGalois/counter mode (GCM), 379–382

authentication and encryption functions, 380

message authentication code, 381Generate function, 218Generator, 100, 121–122Greatest common divisor, 88–89Group master key (GMK), 581Groups, 103

associative, 99closure, 99commutative, 100cyclic, 100distribution, 584finite group, 100generate, 100generator, 100identity element, 100infinite group, 100inverse element, 100keys, 581–582order of, 100

Group temporal key (GTK), 582

HHacking project, 663–664Handshake Protocol, 531–537

action, 533CipherSpec

cipher algorithm, 534CipherType, 534hash size, 535IsExportable, 535IV size, 535key material, 535MAC algorithm, 534

CipherSuite parameteranonymous Diffie-Hellman, 534ephemeral Diffie-Hellman, 534fixed Diffie-Hellman, 534Fortezza, 534RSA, 534

client authentication and key exchange, 536–537

726 Index

Handshake Protocol (Continued)certificate message, 536ephemeral or anonymous

Diffie-Hellman, 536fixed Diffie-Hellman, 536Fortezza, 536RSA, 536

finished message, 537security capabilities, 532–535

cipher suite, 532compression method, 532random, 532session ID, 532–533version, 532

server authentication and key exchange, 535–536

anonymous Diffie-Hellman, 535ephemeral Diffie-Hellman, 535Fortezza, 535–536RSA key exchange, 535

Hardware fault-based attack, 272Hash code, 327Hash function, 314, 358

attack against, 316based on cipher block chaining,

328–329birthday attack, 328meet-in-the-middle-attack, 329

and message authentication codes, 210resistance properties, 324

Hash value, 325Hill cipher, 41–44

algorithm, 42–44concepts from linear algebra, 41–42determinant, 41

HMAC. See also MAC s based on hash functions (HMAC)

algorithm, 369–372design objectives, 369structure, 370

HtE: Hash-then-encrypt, 376HTTPS (HTTP over SSL), 543–544

connection closure, 544connection initiation, 544

Hybrid cloud, 509

IIdeal block cipher, 66Identification payload, 656Identification string exchange, 548Identity and access management (IAM),

518–519Identity element, 100, 295Identity federation, 480–484

examples, 483–484standards, 482–483

SAML, 482SOAP, 482WS-Security, 482XML, 482

Identity management systemadministrators, 479attribute service, 479data consumers, 479elements of

accounting, 479authentication, 478–479authorization, 479delegated administration, 479federation, 479password synchronization, 479provisioning, 479self-service password reset, 479workflow automation, 479

identity provider, 479principal, 479

IEEE 802.11i wireless LAN security, 572–586

authentication phase, 578–580access control approach, 578–579

EAP exchange, 579–580MPDU exchange, 579

discovery phase, 576–578MPDU exchange, 577–578security capabilities, 576

elements of, 574key management phase, 580–584

group keys, 581–582group key distribution, 584pairwise keys, 580–581pairwise key distribution, 582–584

phases of operation, 574–576authentication, 575connection termination, 576discovery, 575key generation and distribution, 576protected data transfer, 576

protected data transfer phase, 584–585CCMP, 584–585TKIP, 584

pseudorandom function, 585–586services, 573

access control, 573authentication, 573privacy with message

integrity, 573IEEE 802.11 wireless LAN overview,

566–572association-related services, 572

association, 572BSS transition, 572disassociation, 572ESS transition, 572no transition, 572reassociation, 572

distribution of messages within a DS, 571

MPDU format, 569network components and architectural

model, 569–570extended service set, 570

protocol architecture, 567–569logical link control, 569media access control, 568–569physical layer, 568

services, 570–572association-related services, 572distribution of messages within a

DS, 571terminology, 567Wi-Fi alliance, 567

IEEE 802.1X Port-Based NAC, 498, 503–506

access control, 505EAPOL (EAP over LAN), 503–505terminology, 504

IKEv2 Exchanges, 652–653Independent BSS (IBSS), 570Indeterminate, 106Index, 247Infinite field, 102Infinite group, 100Infinite ring, 101Informational exchange, 653Infrastructure as a service (IaaS), 508Initialization value (IV), 639Inputs, 215

for single AES round, 148Integral domain, 101, 103Integration, 571Integrity, 10, 12

data, 10system, 10

Intel digital random number generator, 224–225

logical structure, 226processor chip with, 225

Internet Architecture Board (IAB), 628Internet key exchange (IKE)

header and payload formats, 653–657

header format, 653–654key determination protocol, 650–653payload types, 654–657

Internet Key Exchange (IKEv2) Protocol, 630

Internet Security Association and Key Management Protocol (ISAKMP), 649

Intruders, 3, 23–24Intrusion detection, 320Inverse add round key

transformation, 147Inverse element, 100, 295Inverse mix column transformation

(InvMixColumns), 145Inverse shift row transformation

(InvShiftRows), 143Inverse substitute byte transformation

(InvSubBytes), 142Invisible ink, 53Iota step function, 350IP security (IPsec), 626–659

applications, 628–629architecture, 633association database, 633–634

AH Information, 634Anti-Replay Window, 634ESP Information, 634IPsec Protocol Mode, 634Lifetime of this Security

Association, 634Path MTU, 634Security Parameter Index, 634Sequence Counter Overflow, 634Sequence Number Counter, 634

authentication plus confidentiality, 646–647

benefits, 629–630combinations of security associations,

647–649cryptographic suites, 657–659destination address, 633documents, 630–631

architecture, 630authentication header (AH), 630cryptographic algorithms, 631encapsulating security payload

(ESP), 630Internet key exchange (IKE), 630

ESP, 638–644IKE, 650–657policy database, 634–636Protocol mode, 634routing applications, 630scenario, 628–629security associations, 633

IP destination address, 633Security Parameters Index

(SPI), 633Security Protocol Identifier, 633

services, 631traffic processing, 636–638

inbound packets, 637–638outbound packets, 636–637

transport and tunnel modes, 631–632IPv4, 628IPv6, 628Irreducible polynomial, 109, 131Irreversible mapping, 64IS-Box, 140Iteration function, 339

KKeccak, 339Kerberos, 458–475

environmental shortcomingsauthentication forwarding, 471encryption system dependence,

470–471Internet protocol dependence, 471

Index 727

interrealm authentication, 471message byte ordering, 471ticket lifetime, 471

exchanges, 466motivation, 459–460

reliable, 459scalable, 459secure, 459transparent, 459

principal, 468realms and multiple Kerberi, 468–469technical deficiencies

double encryption, 471password attacks, 472PCBC encryption, 471–472session keys, 472

version 4, 460–469authentication dialogue, 463–466secure authentication dialogue,

461–463simple authentication dialogue,

460–461version 5, 469–475

authentication dialogue, 472–474and 4, differences between,

470–472ticket flags, 474–475

Kerberos realm, 468–469Key, 63, 215

determination protocol, 650–653expansion algorithm, 148generation, 271–272length, 217schedule algorithm, 80size, 70unwrapping, 385–387

Key distribution center (KDC), 421, 454Key distribution, symmetric

using asymmetric encryption, 427–430hybrid scheme, 430secret key distribution, 428–430simple secret key distribution,

427–428using symmetric encryption, 418–427

controlling key usage, 425–427decentralized key control, 424–425hierarchical key control, 422–423key distribution scenario, 421–422session key lifetime, 423transparent key control scheme,

423–424Keyed hash function, 318Key encryption key (KEK), 382Key exchange, 262, 549

payload, 656protocols, 290

Key management and distribution, 417–445

distribution of public keys, 430–435hierarchy, 420public-key infrastructure, 443–445symmetric key distribution

using asymmetric encryption, 427–430

using symmetric encryption, 418–427X.509 certificates, 435–443

Key management key, 488Keystream, 219Key wrap mode, 382Key wrapping

algorithm, 383–384KEK, 382and unwrapping, 385–387

Known-plaintext, 32attack on triple DES, 179

LLanes, 343Linear algebra and matrix functionality,

669–670

Linear congruential generators, 210–211Local forwarding, 554Logical link control (LLC), 569

MMAC protocol data unit (MPDU), 568

CRC, 569destination MAC address, 569exchange, 577–579

AS, 579association, 578EAP exchange, 579network and security capability

discovery, 577open system authentication,

577–578secure key delivery, 579

MAC Control, 569MAC header, 569MAC service data unit, 569MAC trailer, 569source MAC address, 569

MAC s based on hash functions (HMAC), 368–372

algorithm, 369–372design objectives, 369efficient implementation, 371security of, 372structure, 370

MAC service data unit (MSDU), 568Mail Delivery Agent (MDA), 616Mail Submission Agent (MSA), 616Management information base (MIB)

content, 631Man-in-the-middle attack, 290–291,

428–429Mask generation function (MGF),

407–408Masquerade, 16, 357Master key, 421, 427Master Secret, 527, 538

Diffie-Hellman, 538RSA, 538

Master session key (MSK), 580Mathematical attacks, 272Maurer’s universal statistical test, 208MD4, 330MD5, 327, 339Media access control (MAC), 568–569Media gateway, 497Meet-in-the-middle attack, 177, 329Message authentication, 315–319

attack against hash function, 316functions, 357–364

hash function, 358MAC, 358, 362–364message encryption, 358–362

keyed hash function, 318message digest, 316requirements, 357

content modification, 357destination repudiation, 357disclosure, 357masquerade, 357sequence modification, 357source repudiation, 357timing modification, 357traffic analysis, 357

simplified examples, 317Message authentication code (MAC),

318, 356, 358, 362–364basic uses of, 363cryptographic checksum, 362requirements for, 365–367

Message digest, 316Message encryption, 358–362

basic uses of message encryption, 359internal and external error

control, 360public-key encryption, 361–362

symmetric encryption, 358–361external error control, 360internal error control, 360

TCP segment, 361Message integrity code (MIC), 583–584Message Store (MS), 616Message Transfer Agent (MTA), 615–616Message User Agent (MUA), 615–616Micali-Schnorr PRNG, 307Michael, 584Miller-Rabin algorithm, 239–241MIPS, 273–274MixColumns, 132, 135

transformation, 144–147Mobile device security, 562–566

strategy, 564–565barrier security, 566device security, 565–566elements, 565traffic security, 566

threats, 563–564interaction with other systems, 564lack of physical security controls,

563–564location services, 564by unknown parties, 564untrusted content, 564untrusted mobile devices, 564untrusted networks, 564

Modification of messages, 16Modular arithmetic, 91–99

congruent modulo, 91Euclidean algorithm

extended, 97–99revisited, 96–99

modulus, 91properties of, 94–96

reducing k modulo n, 94set of residues, or residue classes, 94

Modular polynomial arithmetic, 114–116Modulo operator, 103–104Modulus, 91, 127Monic polynomial, 106Monoalphabetic ciphers, 36–39

digrams, 37permutation, 36relative frequency, 37substitution, 37

MSDU delivery, 571MtE: MAC-then-encrypt, 376Multi-instance model, 515Multiple encryption, 175–180

double DES, 175–177triple DES with three keys, 180triple DES with two keys, 177–180

Multiplication, 101–102, 119Multiplicative identity, 101Multiplicative inverse, 102Multipurpose Internet Mail Extensions

(MIME), 600–606canonical form, 606content types, 601–604

application type, 604message/external-body

subtype, 603message/partial subtype, 603message/rfc822 subtype, 603message type, 603multipart/alternative subtype, 603multipart/digest subtype, 603multipart/mixed subtype, 602multipart/parallel subtype, 602multipart type, 601text type, 601

elements, 600–601example, 604, 605header fields, 601transfer encodings, 604

base64, 604quoted-printable, 604

728 Index

Multirate padding, 340Multi-tenant model, 515Mutual authentication, 453–457, 476–477

NNational Institute of Standards and

Technology (NIST), 72, 130Network access control (NAC), 495–499

context, 497elements of, 496–498

access requestor (AR), 496network access server (NAS), 497policy server, 497

enforcement methods, 498–499DHCP management, 499firewall, 499IEEE 802.1X, 498virtual local area networks

(VLANs), 498–499Network access server (NAS), 497Network security, 8, 520. See also

Cryptographyaccess model, 23basic tasks, 23model for, 22–23secret information, 22security-related transformation, 22threats

information access, 23service, 23

Next-bit test, 212NIST CTR_DRBG, 216–219

entropy source, 217functions, 218generate, 218initialize, 217key length, 217output block length, 217parameters, 217reseed_interval, 217, 219seed length, 217update, 218–219

NIST DSA, 400–404Nonce, 184, 421, 473, 652, 656Nonrepudiation, 19Nonsingular mapping, 64Notarization, 21Notify payload, 656No zero divisors, 101Number of rounds, 70, 79Number theory, 691–696

and finite fields, 677–683

OOakley Key Determination Protocol, 649One-time pad, 47–49One-way authentication, 454, 457, 477One-way function, 263One-way hash function, 263One-way password file, 320Open Shortest Path First (OSPF), 630Optimal asymmetric encryption padding

(OAEP), 277Order, 242, 245Order of group, 100Ordinary polynomial arithmetic, 106–107OSI security architecture

ITU-T3 Recommendation X.800, 14security attack, 14security mechanism, 14security service, 14threats and attacks, 14

Output, 216Output block length, 217Output feedback mode (OFB), 187–189

PPairwise master key (PMK), 580Pairwise transient key (PTK), 580Parameters, 343, 409–410

Passive attack, 15–16release of message contents, 16traffic analysis, 16

Path MTU, 634Perfect secrecy, 48Permutation, 67, 69Permuted input, 74Personal identification number (PIN),

53, 452Personal identity verification (PIV),

484–490algorithms and key sizes, 488authentication, 488–490card application administration

key, 488card issuance and management

subsystem, 485credentials and keys, 487–488documentation, 486–487FIPS 201 PIV system model, 486front-end subsystem, 485system model, 485–486

Pi step function, 348–349PKI, 489Plaintext, 28, 257Platform as a service (PaaS), 508Playfair cipher, 39–41

monarchy, 39plaintext, 40relative frequency of letters, 40

Point at infinity or zero point, 296Policy server, 497Polyalphabetic ciphers, 44–47

autokey system, 46one-time pad, 47–48polyalphabetic substitution

cipher, 44Vernam cipher, 46–47Vigenère cipher, 44–46

Polynomial, 106arithmetic, 106–123

addition, 119coefficient set, 106with coefficients in Zp, 107–109divisor, 109factor, 109greatest common divisor, 110–111indeterminate, 106irreducible, 109modular, 114–116multiplication, 119multiplicative inverse, 116–119using generator, 121–122

constant, 106monic, 106prime, 109ring, 107

Port, 553Powers of an integer, modulo n,

245–246Preimage, 322Preimage resistant, 323Preoutput, 74Pre-shared key (PSK), 580Pretty Good Privacy (PGP), 591–598

cryptographic functions, 594notation, 592–593operational description, 593–598

authentication, 593–595compression, 596–597confidentiality, 595–596confidentiality and

authentication, 596e-mail compatibility, 597–598

services, 593Primality, testing for, 239–242

algorithm, 242distribution of primes, 242Miller-Rabin algorithm, 239–241

details of, 240–241

repeated use of, 241two properties of prime

numbers, 240Prime curve, 299Prime numbers, 232–235Prime polynomial, 109Primitive root, 246Privacy, 10Private cloud, 508Private key, 257–258

efficient operation using, 270–271Product cipher, 66Product systems, 31Programming projects, 665Propagating cipher block chaining

(PCBC), 471Pseudorandom function (PRF), 206,

320, 387Pseudorandomness, 324Pseudorandom number generator

(PRNG), 206, 210–213, 216, 320, 387, 689–691

ANSI X9.17 PRNG, 215–216Blum Blum Shub Generator, 212–213CSPRBG, 212on elliptic curve cryptography,

308–309on hash function, 387–388linear congruential generators, 210–211on MAC function, 389mechanisms based on block

ciphers, 214next-bit test, 212NIST CTR_DRBG, 216–219principles of, 203–210

algorithm design, 209–210requirements, 207–209TRNGs, PRNGs, and PRFs, 205–207use of random numbers, 204–205

pseudorandom function (PRF), 387randomness

consistency, 207frequency test, 208Maurer’s universal statistical

test, 208runs test, 208scalability, 207uniformity, 207

requirements, 207–209on RSA, 307–308

Micali-Schnorr PRNG, 307seed requirements, 208–209unpredictability

backward unpredictability, 208forward unpredictability, 208

using block cipher, 213–219Public cloud, 508Public keys, 257–258, 430–435

(asymmetric) cryptographic algorithm, 255

authority, 432–433certificates, 433–436cryptanalysis, 264distribution scenario, 433public announcement of, 431publication, 432publicly available directory, 431–432uncontrolled distribution, 431

Public key certificate, 255Public-key cryptography, 254, 256–261,

696–704applications for, 262–264

digital signature, 262encryption/decryption, 262key exchange, 262

authentication, 260and secrecy, 261

ciphertext, 258conventional and public-key

encryption, 257, 259

Index 729

decryption algorithm, 258digital signature, 260encryption algorithm, 258encryption with public/private key, 257misconception, 254–255plaintext, 257principles of, 256–264public and private keys, 258public-key cryptanalysis, 264requirements for, 262–264

applications, 262one-way function, 263public-key cryptosystems, 262trap-door one-way function, 263

secrecy, 259secret key, 258

Public key directory, 431–432Public-key encryption, 257, 259Public key infrastructure (PKI), 255Public-key infrastructure X.509 (PKIX),

443–445elements, 255, 443–445

certification authority (CA), 443CRL issuer, 444end entity, 443registration authority (RA), 443repository, 444

management functions, 444–445certification, 445cross certification, 445initialization, 444key pair recovery, 445key pair update, 445registration, 444revocation request, 445

management protocols, 445Purpose-built algorithms, 209Puzzle for Inspector Morse, 53

QQuoted-printable transfer encoding, 604

RRadix-64 encoding, see Base64 transfer

encodingRail fence cipher, 49Randomness, 204Random number generators, 206Random numbers, 204–205

randomness, 204independence, 204uniform distribution, 204

unpredictability, 205RC4, 221–223

initialization of S, 221stream generation, 222strength of, 223strength of RC4, 223

Read-only memory (ROM), 484Realm, 468–470, 473Receiver, 640Relatively prime, 88Relying subsystem, 485Remote access server (RAS), 497Remote forwarding, 555Replay, 16Replay attacks, 453, 640Research projects, 664–665Reseed interval, 217, 219Residue, 94, 127Reversible mapping, 64Rho step function, 347–348Rijndael, 143, 150, 157, 159Rings, 100, 103

abelian group, 101associativity of multiplication, 101closure under multiplication, 101commutativity of multiplication, 101distributive laws, 101integral domain, 101

multiplicative identity, 101no zero divisors, 101

Rivest-Shamir-Adleman (RSA) algorithm, 255, 264–278, 696–699

computational aspects, 267–272exponentiation in modular

arithmetic, 268–269key generation, 271–272private key, 270–271processing of multiple blocks, 268public key, 269–270

processeing of multiple blocks, 268

RSA-PSS digital signature algorithm, 407–412

mask generation function (MGF), 407–408

message encoding, 408–410RSA-PSS EM verification, 411signature verification, 410–412

security of, 272–278brute force, 272CCA, 277chosen ciphertext attacks, 272ciphertext attack and asymmetric

encryption padding, 277factoring problem, 272–275fault-based attack, 276hardware fault-based attack, 272mathematical attacks, 272MIPS-years needed to factor, 274OAEP, 277progress in factorization, 273timing attacks, 272, 275–276

Robust Security Network (RSN), 573Rotor machines, 50–52

DES, 52multiple cylinders, 52single-cylinder system, 52with wiring by numbered contacts, 51

Round, 68, 75–76constants in SHA-3, 350function, 68, 70

Routing control, 21Runs test, 208

SSage computer algebra projects, 662–663S-Box, 138, 140, 171–173

nibble substitution, 168Scalability, 207Schnorr digital signature scheme,

398, 400Second assertion, 243Second preimage resistant, 323Secret-key encryption, 29, 31, 258Secure Hash Algorithm (SHA), 329–339Secure shell (SSH), 544–555

Connection Protocol, 551–555Transport Layer Protocol, 545–550User Authentication Protocol, 550–551

Secure sockets layer (SSL), 525–538Alert Protocol, 530–531architecture, 526–527Change Cipher Spec Protocol, 530, 531connection state, 526, 527

Cipher spec, 527compression method, 527is resumable, 527master secret, 527peer certificate, 527session identifier, 527

cryptographic computations, 537–538generation of cryptographic

parameters, 538master secret creation, 538

Handshake Protocol, 531–537Record Protocol, 527–530

compression, 528confidentiality, 527

fragmentation, 528header, fields of, 530MAC, 528–529message integrity, 527–529

session state, 526, 527client write key, 527client write MAC secret, 527initialization vectors, 527sequence numbers, 527server and client random, 527server write key, 527server write MAC secret, 527

Security as a service (SecaaS), 517Security Assertion Markup Language

(SAML), 482Security association database (SAD),

632, 633–634AH information, 634Anti-Replay Window, 634ESP information, 634IPsec Protocol Mode, 634Lifetime of this Security

Association, 634Path MTU, 634Security Parameter Index, 634Sequence Counter Overflow, 634Sequence Number Counter, 634

Security associations (SA), 633authentication plus confidentiality,

646–647ESP with authentication option, 646transport adjacency, 646–647transport-tunnel bundle, 647

combinations of, 647–649IP Destination Address, 633payload

attribute, 655proposal, 655transform, 655

Security Parameters Index (SPI), 633Security Protocol Identifier, 633

Security attacks, 14–17active attacks, 16–17

denial of service, 16masquerade, 16modification of messages, 16replay, 16

passive attacks, 15–16release of message contents, 16traffic analysis, 16

Security audit trail, 20Security information and event manage-

ment (SIEM), 519“Security in the Internet Architecture”

(RFC 1636), 628Security label, 20Security mechanisms

for cryptographic hash functions, 323of elliptic curve cryptography, 306of HMAC, 372of MACs, 367–368

brute-force attacks, 367–368computation resistance, 367cryptanalysis, 368

pervasiveauthentication exchange, 21event detection, 20security audit trail, 20security label, 20security recovery, 20trusted functionality, 20

recovery, 20of RSA, 272–278services, 17–20

access control, 18availability service, 19–20data confidentiality, 19data integrity, 19nonrepudiation, 19

and services, relationship between, 21

730 Index

Security mechanisms (Continued)specific

access control, 20authentication exchange, 21data integrity, 20digital signature, 20encipherment, 20notarization, 21routing control, 21traffic padding, 21

Security Parameters Index (SPI), 633–634

Security policy database (SPD), 632, 634–636

local and remote ports, 635local IP address, 635name, 635next layer protocol, 635remote IP address, 635

Security Protocol Identifier, 633Seed, 206Seed length, 217Seed requirements, 208–209Selectors, 635Sender, 640Sequence counter overflow, 634Sequence number counter, 634Session key, 419Session security module (SSM), 423Set of residues, 94SHA-0, 329SHA-1, 329SHA-2, 329SHA-3, 339–350

iteration function f, 343–350Chi step function, 349–350composition, 344Iota step function, 350Pi step function, 348–349Rho step function, 347–348round constants in SHA-3, 350state matrix, 344step functions in SHA-3, 345theta step function, 345–347

parameters, 343sponge construction, 339–343

absorbing phase, 342bitrate, 339capacity, 341iteration function, 339multirate padding, 340simple padding, 340sponge function input and

output, 340squeezing phase, 342

SHA-224, 330SHA-256, 330SHA-384, 330SHA-512, 330

logic, 337big-endian, 331message digest generation using

SHA-512, 331constants, 333processing of single 1024-bit

block, 332step 1 append padding bits, 330step 2 append length, 330step 3 initialize hash buffer, 331step 4 process message in 1024-bit

(128-word) blocks, 331step 5 output, 332

round function, 334–336ShiftRows, 132, 135

AES row and column operations, 144

forward shift row transformation (ShiftRows), 143

inverse shift row transformation (InvShiftRows), 143

Signing Domain IDentifier (SDID), 621Simple Mail Transfer Protocol

(SMTP), 454Simple Object Access Protocol

(SOAP), 482Simple padding, 340Simplified AES (S-AES)

encryption and decryption, 166–171structure, 172–173

Single-cylinder system, 52Single-key encryption, 31Skew, 224

deskewing algorithms, 224S/MIME, 599–615

certificate processing, 612–614cryptographic algorithms

MUST, 607SHOULD, 607–608

enhanced security services, 614–615secure mailing lists, 615security labels, 614signed receipts, 614

functionality, 606–608clear-signed data, 606–607enveloped data, 606signed and enveloped data, 607signed data, 606

messages, 608–609certificates-only message, 612envelopedData, 609–610registration request, 612securing a MIME entity, 609signedData, 610–612

MIME, 600–606RFC 5322, 599–600user agent role, 612

certificate storage and retrieval, 612key generation, 612registration, 612

VeriSign certificates, 613, 614Software as a service (SaaS), 508Sound/video input, 223Sponge construction, 339–343

absorbing phase, 342bitrate, 339capacity, 341iteration function, 339multirate padding, 340simple padding, 340sponge function input and

output, 340squeezing phase, 342

Sponge function input and output, 340

Squeezing phase, 342State, 132State array, 132State matrix, 344Static biometrics, 452Steganography, 52–54

advantage, 54character marking, 53drawbacks, 54invisible ink, 53pin punctures, 53typewriter correction ribbon, 53

Stream ciphers, 31, 63, 689–691keystream, 219

Stream generation, 222Strict avalanche criterion (SAC), 79Strong collision resistance, 323SubBytes, 132, 138Subkey, 70, 74Substitute bytes, 135, 137–143

AES byte-level operations, 138construction of S-Box and IS-Box, 140forward substitute byte transformation

(SubBytes), 138inverse substitute byte transformation

(InvSubBytes), 142

Substitution-permutation network (SPN), 69

Substitution techniques, 33–48, 67–68Caesar cipher, 34–36Hill cipher, 41–44monoalphabetic ciphers, 36–39one-time pad, 47–49Playfair cipher, 39–41polyalphabetic ciphers, 44–47

Supplicant, 496, 501Suppress-replay attacks, 456Symmetric block ciphers, 210Symmetric card authentication

key, 488Symmetric cipher model, 28–33

ciphertext, 29cryptanalysis and brute-force

attack, 31attacks on encrypted messages, 32brute-force attack, 31, 33computationally secure encryption

scheme, 33cryptanalysis, 31unconditionally secure encryption

scheme, 33cryptography, 31

keys used, 31plaintext, processed, 31plaintext to ciphertext, 31

decryption algorithm, 29encryption algorithm, 29model of symmetric

cryptosystem, 30plaintext, 28secret key, 29secure use of conventional

encryption, 29simplified model of symmetric

encryption, 29Symmetric cryptosystem, 30Symmetric encryption, 31, 358–361

external error control, 360internal error control, 360

TTag, 365Temporal Key Integrity Protocol

(TKIP), 584Theta step function, 345–347Ticket, 461, 474–475Ticket-granting service exchange, 462Time complexity, 267–268Timestamp, 398Timing attacks, 78, 272, 275–276Timing complexity, 263, 267Traditional block cipher structure,

63–72block cipher, 63confusion, 67–68diffusion, 67Feistel cipher, 66–70

block size, 70ease of analysis, 70encryption and decryption, 69fast software encryption/

decryption, 70key size, 70number of rounds, 70permutation, 67, 69round function, 68, 70subkey generation algorithm, 70substitution, 67–68substitution-permutation network

(SPN), 69Feistel decryption algorithm, 70–72Feistel example, 72motivation for the Feistel cipher

structure, 63–66arbitrary reversible substitution

cipher, 66

Index 731

encryption and decryption tables for substitution, 65

ideal block cipher, 66reversible or nonsingular, 64

stream cipher, 63Traffic analysis, 16Traffic flow confidentiality (TFC), 639Traffic padding, 21Transformation functions, AES, 137–148

AddRoundKey transformationforward add round key transforma-

tion (AddRoundKey), 147inputs for single AES round, 148inverse add round key transforma-

tion, 147MixColumns transformation, 144–147

forward mix column transformation (MixColumns), 144

inverse mix column transformation (InvMixColumns), 145

ShiftRows transformation, 143–144AES row and column

operations, 144forward shift row transformation

(ShiftRows), 143inverse shift row transformation

(InvShiftRows), 143substitute bytes transformation,

137–143AES byte-level operations, 138construction of S-Box and IS-Box, 140forward substitute byte transformation

(SubBytes), 138inverse substitute byte transformation

(InvSubBytes), 142Transport Layer Protocol, 545–550

cryptographic algorithms, 548host keys, 545–546key generation, 549–550packet exchange, 546–549, 554

algorithm negotiation, 548end of key exchange, 549identification string exchange, 548key exchange, 549MAC, 547–548packet length, 547padding length, 547payload, 547random padding, 547service request, 549

Transport Layer Security (TLS), 522–555alert codes, 541cipher suites, 542client certificate types, 542cryptographic computations, 542–543HTTPS, 543–544message authentication code, 539padding, 543pseudorandom function, 539–541secure shell (SSH), 544–555secure sockets layer (SSL), 525–538version number, 539web security considerations, 523–525

Transport modes, 631–632, 641–644Transposition cipher, 49–50Transposition techniques, 49–50

rail fence technique, 49–50Trap-door one-way function, 263–264Triple DES (3DES)

with three keys, 180

with two keys, 177–180known-plaintext attack on triple

DES, 179True random number generator

(TRNG), 205, 223–227comparison of PRNGs and TRNGs,

224DRNG hardware architecture,

225–227CBC-MAC or CMAC, 225Intel DRNG logical structure, 226Intel processor chip, 225

DRNG logical structure, 227entropy sources, 223

disk drives, 223sound/video input, 223

Intel digital random number genera-tor, 224–225

skew, 224deskewing algorithms, 224

Trust, 2–3Trusted functionality, 20Tunnel modes, 551, 631–632, 641–644Tweakable block cipher, 192–193Typewriter correction ribbon, 53

UUnconditionally secure, 33Uniform distribution, 204Uniformity, 207Unpredictability, 205Update function, 218–219User authentication, 450–490

federated identity management, 478–484

Kerberos, 458–475personal identity verification (PIV),

484–490principles, 451–454using asymmetric encryption, 476–478using symmetric encryption, 454–458

User Authentication Protocol, 550–551authentication methods, 551message exchange, 550–551message types and formats, 550

U.S. National Security Agency (NSA), 308

VVendor ID payload, 657Vernam cipher, 46–47Vigenère cipher, 44–46Virtual local area networks (VLANs),

498–499Virtual private network, 641Virus detection, 320

WWeb security, 519

considerations, 523–525threats, 524traffic security approaches, 525

Weierstrass equation, 296Wi-Fi, 559Wi-Fi Protected Access (WPA), 567, 573Wired Equivalent Privacy (WEP),

573, 584Wireless LAN (WLAN)

overview, see IEEE 802.11 wireless LAN overview

security, see IEEE 802.11i wireless LAN security

Wireless network security, 558–586components, 560IEEE 802.11i wireless LAN security,

572–586IEEE 802.11 wireless LAN overview,

566–572measures, 561–562

securing wireless access points, 562securing wireless networks, 562securing wireless transmissions,

561–562mobile device security, 562–566threats, 560–561wireless security, 559–562

XX.509 certificates, 435–443

certificate subject and issuer attri-butes, 442–443

issuer alternative name, 443subject alternative name, 442subject directory attributes, 443

certification authority (CA)forward certificates, 439reverse certificates, 439

certification path constraints, 443basic constraints, 443name constraints, 443policy constraints, 443

formats, 437–438extensions, 438issuer name, 437issuer unique identifier, 437period of validity, 437serial number, 437signature, 438signature algorithm identifier, 437subject name, 437subject’s public-key information, 437subject unique identifier, 438version, 437

hierarchy, 440key and policy information, 442

authority key identifier, 442certificate policies, 442key usage, 442policy mappings, 442private-key usage period, 442subject key identifier, 442

revocation of, 440–441user’s, 438–440Version 3, 441–442

XTS-AES mode, 191–198feedback characteristic of modes of

operation, 192operation on a sector, 196–198

ciphertext-stealing, 196–198operation on a single block, 194–196storage encryption requirements,

193–194tweakable block cipher, 192–193XTS-AES mode, 197

ZZero point, 296ZIP, 36


ACRONYMS

3DES Triple DES

AES Advanced Encryption Standard

AH Authentication Header

ANSI American National Standards Institute

CBC Cipher Block Chaining

CESG Communications-Electronics Security

Group

CFB Cipher Feedback

CMAC Cipher-Based Message Authentication

Code

CRT Chinese Remainder Theorem

DoS Denial of Service

DEA Data Encryption Algorithm

DES Data Encryption Standard

DoS Denial of Service

DSA Digital Signature Algorithm

DSS Digital Signature Standard

ECB Electronic Codebook

ECC Elliptic Curve Cryptography

ECDSA Elliptic Curve Digital Signature

Algorithm

ESP Encapsulating Security Payload

FIPS Federal Information Processing

Standard

IAB Internet Architecture Board

IETF Internet Engineering Task Force

IP Internet Protocol

IPsec IP Security

ISO International Organization for

Standardization

ITU International Telecommunication

Union

ITU-T ITU Telecommunication

Standardization Sector

IV Initialization Vector

KDC Key Distribution Center

LAN Local Area Network

MAC Message Authentication Code

MIC Message Integrity Code

MIME Multipurpose Internet Mail

Extension

MD5 Message Digest, Version 5

MTU Maximum Transmission Unit

NIST National Institute of Standards and

Technology

NSA National Security Agency

OFB Output Feedback

PCBC Propagating Cipher Block Chaining

PGP Pretty Good Privacy

PIV Personal Identity Verification

PKI Public Key Infrastructure

PRNG Pseudorandom Number Generator

RFC Request for Comments

RNG Random Number Generator

RSA Rivest-Shamir-Adelman

RSA-PSS RSA Probabilistic Signature Scheme

SHA Secure Hash Algorithm

S/MIME Secure MIME

SNMP Simple Network Management

Protocol

SNMPv3 Simple Network Management

Protocol Version 3

SSL Secure Sockets Layer

TCP Transmission Control Protocol

TDEA Triple DEA

TLS Transport Layer Security

UDP User Datagram Protocol

WAN Wide Area Network

THE WILLIAM STALLINGS BOOKS ON COMPUTER

DATA AND COMPUTER COMMUNICATIONS, NINTH EDITIONA comprehensive survey that has become the standard in the field, covering (1) data communications, including transmission, media, signal encoding, link control, and multiplexing; (2) communication networks, including circuit- and packet-switching, frame relay, ATM, and LANs; (3) the TCP/IP protocol suite, including IPv6, TCP, MIME, and HTTP, as well as a detailed treatment of network security. Received the 2007 Text and Academic Authors Association (TAA) award for the best Computer Science and Engineering Textbook of the year. ISBN 978-0-13-139205-2

OPERATING SYSTEMS, SEVENTH EDITIONA state-of-the art survey of operating system principles. Covers fundamental technology as well as contemporary design issues, such as threads, microkernels, SMPs, real-time systems, multiprocessor scheduling, embedded OSs, distributed systems, clusters, security, and object-oriented design. Third, fourth, and sixth editions received the TAA award for the best Computer Science and Engineering Textbook of the year. ISBN 978-0-13-230998-1

BUSINESS DATA COMMUNICATIONS, SEVENTH EDITION (with Tom Case)

A comprehensive presentation of data communications and telecommunications from a business perspective. Covers voice, data, image, and video communications and applications technology and includes a number of case studies. Topics covered include data communications, TCP/IP, cloud computing, Internet protocols and applications, LANs and WANs, network security, and network management. ISBN 978-0133023893

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A unified view of this broad field. Covers fundamentals such as CPU, control unit, microprogramming, instruction set, I/O, and memory. Also covers advanced topics such as multicore, superscalar, and parallel organization. Four-time winner of the TAA award for the best Computer Science and Engineering Textbook of the year. ISBN 978-0-13-293633-0

AND DATA COMMUNICATIONS TECHNOLOGY

COMPUTER SECURITY, SECOND EDITION (with Lawrie Brown)A comprehensive treatment of computer security technology, including algorithms, protocols, and applications. Covers cryptography, authentication, access control, database security, intrusion detection and prevention, malicious software, denial of service, firewalls, software security, physical security, human factors, auditing, legal and ethical aspects, and trusted systems. Received the 2008 Text and Academic Authors Association (TAA) award for the best Computer Science and Engineering Textbook of the year. ISBN 978-0-13-277506-9

NETWORK SECURITY ESSENTIALS, FIFTH EDITIONA tutorial and survey on network security technology. The book covers important network security tools and applications, including S/MIME, IP Security, Kerberos, SSL/TLS, SET, and X509v3. In addition, methods for countering hackers and viruses are explored. ISBN: 0-13-337043-7

WIRELESS COMMUNICATIONS AND NETWORKS, SECOND EDITION

A comprehensive, state-of-the art survey. Covers fundamental wireless communications topics, including antennas and propagation, signal encoding techniques, spread spectrum, and error correction techniques. Examines satellite, cellular, wireless local loop networks and wireless LANs, including Bluetooth and 802.11. Covers Mobile IP and WAP. ISBN 0-13-191835-4

COMPUTER NETWORKS WITH INTERNET PROTOCOLS AND TECHNOLOGY

An up-to-date survey of developments in the area of Internet-based protocols and algorithms. Using a top-down approach, this book covers applications, transport layer, Internet QoS, Internet routing, data link layer and computer networks, security, and network management. ISBN 0-13141098-9

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